Not exactly. Let's examine each of the keywords in the passage.
Matthew 10:28 "And fear not them which kill[ἀποκτείνω] the body, but are not able to kill the soul[ψυχή]: but rather fear him which is able to destroy[ἀπόλλυμι] both soul and body in hell."
(1) ἀποκτείνω(kill):
Strong's Concordance -
From apo and kteino (to slay); to kill outright; figuratively, to destroy -- put to death, kill, slay;
Thayer's Greek Lexicon -
ἀποκτείνω, and Aeolic, ἀποκτέννω (Matthew 10:28 L T Tr; Mark 12:5 G L T Tr; Luke 12:4 L T Tr; 2 Corinthians 3:6 T Tr; cf. Fritzsche on Mark, p. 507f; (Tdf. Proleg., p. 79); Winers Grammar, 83 (79); (Buttmann, 61 (54))), ἀποκτενῶ (Griesbach in Matthew 10:28; Luke 12:4), ἀποκταίνω (Lachmann in 2 Corinthians 3:6; Revelation 13:10), ἀποκτεννυντες (Mark 12:5 WH); future ἀποκτενῶ; 1 aorist ἀπέκτεινα; passive, present infinitive ἀποκτέννεσθαι (Revelation 6:11 G L T Tr WH); 1 aorist ἀπεκτάνθην (Alexander Buttmann (1873) Ausf. Spr. ii. 227; Winers Grammar, the passage cited; (Buttmann, 41 (35f))); (from Homer down);
- properly, to kill in any way whatever (ἀπό i.e. so as to put out of the way; cf. (English to kill off), German abschlachten): Matthew 16:21; Matthew 22:6; Mark 6:19; Mark 9:31; John 5:18; John 8:22; Acts 3:15; Revelation 2:13, and very often; (ἀποκτενῶ ἐν θανάτῳ, Revelation 2:23; Revelation 6:8, cf. Buttmann, 184 (159); Winers Grammar, 339 (319)). to destroy (allow to perish): Mark 3:4 (yet others take it here absolutely, to kill).
- metaphorically, to extinguish, abolish: τήν ἐχτραν, Ephesians 2:16; to inflict moral death, Romans 7:11 (see ἀποθνῄσκω, II. 2); to deprive of spiritual life and procure eternal misery, 2 Corinthians 3:6 (Lachmann ἀποκταινει; see above).
So, the word ἀποκτείνω does not solely mean, "to put to death", i.e. to slay. It can likewise mean "to destroy or let perish", i.e. bring to ruin. It can also, at times, be used interchangeably with ἀπόλλυμι(the word translated as destroyed in Matthew 10:28). For example;
Mark 3:4 "And he said to them, "Is it lawful on the Sabbath to do good or to do harm, to save life or to kill[ἀποκτείνω]?" But they were silent."
Luke 6:9 "And Jesus said to them, "I ask you is it lawful on the Sabbath to do good or to do harm, to save life or to destroy[ἀπόλλυμι] it?"
Both Mark 3:4 and Luke 6:9 were referring to the exact same situations, yet different words are used. Thus, ἀπόλλυμι and ἀποκτείνω can, at times, be used in an interchangeable manner; however, they are not identical. ἀπόλλυμι is more intense than ἀποκτείνω.
(2) ἀπόλλυμι(destroy):
Strong's Concordance -
From apo and the base of olethros; to destroy fully (reflexively, to perish, or lose), literally or figuratively -- destroy, die, lose, mar, perish.
Thayer's Greek Lexicon -
ἀπόλλυμι and ἀπολλύω ((ἀπολλύει John 12:25 T Tr WH), imperative ἀπόλλυε Romans 14:15 (cf. Buttmann, 45 (39); WH's Appendix, p. 168f)); future ἀπολέσω and (1 Corinthians 1:19 ἀπολῶ from a passage in the O. T., where often) ἀπολῶ (cf. Winers Grammar, 83 (80); (Buttmann, 64 (56))); 1 aorist ἀπώλεσα; to destroy; middle, present ἀπόλλυμαι; (imperfect 3 person plural ἀπώλλυντο 1 Corinthians 10:9 T Tr WH); future ἀπολοῦμαι; 2 aorist ἀπωλόμην; (2 perfect active participle ἀπολωλώς); (from Homer down); to perish.
- to destroy i.e. to put out of the way entirely, abolish, put an end to, ruin: Mark 1:24; Luke 4:34; Luke 17:27, 29; Jude 1:5; τήν σοφίαν render useless, cause its emptiness to be perceived, 1 Corinthians 1:19 (from the Sept. of Isaiah 29:14); to kill: Matthew 2:13; Matthew 12:14; Mark 9:22; Mark 11:18; John 10:10, etc.; contextually, to declare that one must be put to death: Matthew 27:20; metaphorically, to devote or give over to eternal misery: Matthew 10:28; James 4:12; contextually, by one's conduct to cause another to lose eternal salvation: Romans 14:15. Middle to perish, to be lost, ruined, destroyed;
a. of persons; (a). properly: Matthew 8:25; Luke 13:3, 5, 33; John 11:50; 2 Peter 3:6; Jude 1:11, etc.; ἀπόλλυμαι λιμῷ, Luke 15:17; ἐν μαχαρια, Matthew 26:52; καταβαλλόμενοι, ἀλλ' οὐκ ἀπολλύμενοι, 2 Corinthians 4:9. (b). tropically, to incur the loss of true or eternal life; to be delivered up to eternal misery: John 3:15 (R Lbr.), ; (it must be borne in mind, that according to John's conception eternal life begins on earth, just as soon as one becomes united to Christ by faith); Romans 2:12; 1 Corinthians 8:11; 1 Corinthians 15:18; 2 Peter 3:9. Hence, οἱ σῳζόμενοι they to whom it belongs to partake of salvation, and οἱ ἀπολλύμενοι those to whom it belongs to perish or to be consigned to eternal misery, are contrasted by Paul: 1 Corinthians 1:18; 2 Corinthians 2:15; 2 Corinthians 4:8; 2 Thessalonians 2:10 (on these present participles, cf. Winers Grammar, 342 (321); Buttmann, 206 (178)).
b. of things; to be blotted out, to vanish away: ἡ εὐπρέπεια, James 1:11; the heavens, Hebrews 1:11 (from Psalm 101:27 (); to perish — "of things which on being thrown away are decomposed, as μέλος τοῦ σώματος, Matthew 5:29f; remnants of bread, John 6:12; — or which perish in some other way, as βρῶσις, John 6:27; χρυσίον, 1 Peter 1:7; — or which are mined so that they can no longer subserve the use for which they were designed, as οἱ ἀσκοί: Matthew 9:17; Mark 2:22; Luke 5:37.
- to destroy i.e. to lose;
a. properly: Matthew 10:42; Mark 9:41 (τόν μισθόν αὐτοῦ); Luke 15:4, 8, 9; Luke 9:25; Luke 17:33; John 12:25; 2 John 1:8, etc.
b. metaphorically, Christ is said to lose anyone of his followers (whom the Father has drawn to discipleship) if such a one becomes wicked and fails of salvation: John 6:39, cf. John 18:9. Middle to be lost: θρίξ ἐκ τῆς κεφαλῆς, Luke 21:18; θρίξ ἀπό τῆς κεφαλῆς, Acts 27:34 (Rec. πεσεῖται); τά λαμπρά ἀπώλετο ἀπό σου, Revelation 18:14 (Rec. ἀπῆλθε). Used of sheep, straying from the flock: properly, Luke 15:4 (τό ἀπολωλός, in Matthew 18:12 τό πλανώμενον). Metaphorically, in accordance with the O. T. comparison of the people of Israel to a flock (Jeremiah 27:6 (); Ezekiel 34:4, 16), the Jews, neglected by their religious teachers, left to themselves and thereby in danger of losing eternal salvation, wandering about as it were without guidance, are called τά πρόβατα τά ἀπολωλότα τοῦ οἴκου Ἰσραήλ: Matthew 10:6; Matthew 15:24 (Isaiah 53:6; 1 Peter 2:25); and Christ, reclaiming them from wickedness, is likened to a shepherd and is said ζητεῖν καί σῴζειν τό ἀπολωλός: Luke 19:10; Matthew 18:11 Rec. (Compare: συναπόλλυμι.)
ἀπόλλυμι unequivocally has a more intense connotation to it than ἀποκτείνω.
(3) ψυχή(soul):
Strong's Concordance -
From psucho; breath, i.e. (by implication) spirit, abstractly or concretely (the animal sentient principle only; thus distinguished on the one hand from pneuma, which is the rational and immortal soul; and on the other from zoe, which is mere vitality, even of plants: these terms thus exactly correspond respectively to the Hebrew nephesh, ruwach and chay) -- heart (+ -ily), life, mind, soul, + us, + you.
Thayer's Greek Lexicon -
ψυχή, ψυχῆς, ἡ (ψύχω, to breathe, blow), from Homer down, the Sept. times too many to count for נֶפֶשׁ, occasionally also for לֵב and לֵבָב;
- breath (Latinanima), i.e.
a. the breath of life; the vital force which animates the body and shows itself in breathing: Acts 20:10; of animals, Revelation 8:9 (Genesis 9:4; Genesis 35:18; ἐπιστραφήτω ψυχή τοῦ παιδαρίου, 1 Kings 17:21); so also in those passages where, in accordance with the trichotomy or threefold division of human nature by the Greeks, ἡ ψυχή; is distinguished from τό πνεῦμα (see πνευαμ, 2, p. 520a (and references under the word πνεῦμα 5)), 1 Thessalonians 5:23; Hebrews 4:12.
b. life: μέριμναν τῇ ψυχή, Matthew 6:25; Luke 12:22; τήν ψυχήν ἀγαπᾶν, Revelation 12:11; (μισεῖν, Luke 14:26); τιθέναι, John 10:11, 15, 17; John 13:37; John 15:13; 1 John 3:16; παραδιδόναι, Acts 15:26; διδόναι (λύτρον, which see), Matthew 20:28; Mark 10:45; ζητεῖν τήν ψυχήν τίνος (see ζητέω, 1 a.), Matthew 2:20; Romans 11:3; add, Matthew 6:25; Mark 3:4; Luke 6:9; Luke 12:20, 23; Acts 20:24; Acts 27:10, 22; Romans 16:4; 2 Corinthians 1:23; Philippians 2:30; 1 Thessalonians 2:8; in the pointed aphorisms of Christ, intended to fix themselves in the minds of his hearers, the phrases εὑρίσκειν, σῴζειν, ἀπολλύναι τήν ψυχήν αὐτοῦ, etc., designate as ψυχή in one of the antithetic members the life which is lived on earth, in the other, the (blessed) life in the eternal kingdom of God: Matthew 10:39; Matthew 16:25; Mark 8:35-37; Luke 9:24, 56 Rec.; ; John 12:25; the life destined to enjoy the Messianic salvation is meant also in the following phrases ((where R. V. soul)): περιποίησις ψυχῆς, Hebrews 10:39; κτᾶσθαι τάς ψυχάς, Luke 21:19; ὑπέρ τῶν ψυχῶν (here A. V. (not R. V.) for you; cf.
c. below), 2 Corinthians 12:15.
c. that in which there is life; a living being: ψυχή ζῶσα, a living soul, 1 Corinthians 15:45; (Revelation 16:3 R Tr marginal reading) (Genesis 2:7; plural ); πᾶσα ψυχή ζωῆς, Revelation 16:3 (G L T Tr text WH) (Leviticus 11:10); πᾶσα ψυχή, every soul, i.e. everyone, Acts 2:43; Acts 3:23; Romans 13:1 (so כָּל־נֶפֶשׁ, Leviticus 7:17 (27); ); with ἀνθρώπου added, every soul of man (אָדָם נֶפֶשׁ, Numbers 31:40, 46 (cf. 1 Macc. 2:38)), Romans 2:9. ψυχαί, souls (like the Latincapita) i.e. persons (in enumerations; cf. German Seelenzahl): Acts 2:41; Acts 7:14; Acts 27:37; 1 Peter 3:20 (Genesis 46:15, 18, 22, 26, 27; Exodus 1:5; Exodus 12:4; Leviticus 2:1; Numbers 19:11, 13, 18; (Deuteronomy 10:22); the examples from Greek authors (cf. Passow, under the word, 2, vol. ii, p. 2590b) are of a different sort (yet cf. Liddell and Scott, under the word, II. 2)); ψυχαί ἀνθρώπων of slaves (A. V. souls of men (R. V. with marginal reading 'Or lives')), Revelation 18:13 (so (Numbers 31:35); Ezekiel 27:13; see σῶμα, 1 c. (cf. Winer's Grammar, § 22, 7 N. 3)).
- the soul (Latinanimus), a. the seat of the feelings, desires, affections, aversions (our soul, heart, etc. (R. V. almost uniformly soul); for examples from Greek writings see Passow, under the word, 2, vol. ii., p. 2589b; (Liddell and Scott, under the word, II. 3); Hebrew נֶפֶשׁ, cf. Gesenius, Thesaurus ii, p. 901 in 3): Luke 1:46; Luke 2:35; John 10:24 (cf. αἴρω, 1 b.); Acts 14:2, 22; Acts 15:24; Hebrews 6:19; 2 Peter 2:8, 14; ἡ ἐπιθυμία τῆς ψυχῆς, Revelation 18:14; ἀνάπαυσιν ταῖς ψυχαῖς εὑρίσκειν, Matthew 11:29; ψυχή, ... ἀναπαύου, φάγε, πίε (WH brackets these three imperatives), εὐφραίνου (personification and direct address), Luke 12:19, cf. Luke 12:18 (ἡ ψυχή ἀναπαύσεται, Xenophon, Cyril 6, 2, 28; ἐυφραίνειν τήν ψυχήν, Aelian v. h. 1, 32); εὐδοκεῖ ἡ ψυχή μου (anthropopathically, of God), Matthew 12:18; Hebrews 10:38; περίλυπος ἐστιν ἡ ψυχή μου, Matthew 26:38; Mark 14:34; ἡ ψυχή μου τετάρακται, John 12:27; ταῖς ψυχαῖς ὑμῶν ἀκλυόμενοι (fainting in your souls (cf. ἐκλύω, 2 b.)), Hebrews 12:3; ἐν ὅλῃ τῇ ψυχή σου, with all thy soul, Matthew 22:37; (Luke 10:27 L text T Tr WH); ἐξ ὅλης τῆς ψυχῆς σου (Latinex toto animo), with (literally, from (cf. ἐκ, II. 12 b.)) all thy soul, Mark 12:30, 33 (here T WH omit; L Tr marginal reading brackets the phrase); Luke 10:27 (R G) (Deuteronomy 6:5; (Epictetus diss. 3, 22, 18 (cf. Xenophon, anab. 7, 7, 43)); Antoninus 3, 4; (especially 4, 31; 12, 29); ὅλῃ τῇ ψυχή φροντίζειν τίνος (rather, with κεχαρισθαι), Xenophon, mem. 3, 11, 10); μία ψυχή, with one soul (cf. πνεῦμα, 2, p. 520a bottom), Philippians 1:27; τοῦ πλήθους ... ἦν ἡ καρδία καί ἡ ψυχή μία, Acts 4:32 (ἐρωτηθεις τί ἐστι φίλος, ἔφη. μία ψυχή δύο σώμασιν ἐνοικουσα, (Diogenes Laërtius 5, 20 (cf. Aristotle, eth. Nic. 9, 8, 2, p. 1168b, 7; on the elliptical ἀπό μιᾶς (namely, ψυχῆς?), see ἀπό, III.)); ἐκ ψυχῆς, from the heart, heartily (Ephesians 6:6 (Tr WH with Ephesians 6:7)); Colossians 3:23 (ἐκ τῆς ψυχῆς often in Xenophon; τό ἐκ ψυχῆς πένθος, Josephus, Antiquities 17, 6, 5).
b. "the (human) soul in so far as it is so constituted that by the right use of the aids offered it by God it can attain its highest end and secure eternal blessedness, the soul regarded as a moral being designed for everlasting life": 3 John 1:2; ἀγρύπνειν ὑπέρ τῶν ψυχῶν, Hebrews 13:17; ἐπιθυμίαι, αἵτινες στρατεύονται κατά τῆς ψυχῆς, 1 Peter 2:11; ἐπίσκοπος τῶν ψυχῶν, 1 Peter 2:25; σῴζειν τάς ψυχάς, James 1:21; ψυχήν ἐκ θανάτου, from eternal death, James 5:20; σωτηρία ψυχῶν, 1 Peter 1:9; ἁγνίζειν τάς ψυχάς ἑαυτῶν, 1 Peter 1:22; (τάς ψυχάς πιστῷ κτίστῃ παρατίθεσθαι, 1 Peter 4:19).
c. the soul as an essence which differs from the body and is not dissolved by death (distinguished from τό σῶμα, as the other part of human nature (so in Greek writings from Isocrates and Xenophon down; cf. examples in Passow, under the word, p. 2589{a} bottom; Liddell and Scott, under the word, II. 2)): Matthew 10:28, cf. 4 Macc. 13:14 (it is called ἀθάνατος, Herodotus 2, 123; Plato Phaedr., p. 245 c., 246 a., others; ἄφθαρτος, Josephus, b. j. 2, 8, 14; διαλυθῆναι τήν ψυχήν ἀπό τοῦ σώματος, Epictetus diss. 3, 10, 14); the soul freed from the body, a disembodied soul, Acts 2:27, 31 Rec.; Revelation 6:9; Revelation 20:4 (Wis. 3:1; (on the Homeric use of the word, see Ebeling, Lex. Homer, under the word, 3, and references at the end, also Proudfit in Bib. Sacr. for 1858, pp. 753-805)).
As you can see, the word ψυχή does not solely refer to one's inner consciousness or mind(soul). It also refers to the life force, the breath of life, or vital spirit that sustains life(and consequently leads to sentience, i.e. the ability to feel or perceive things, belonging to both humans and animals), according to Strong's and Thayer's. This is due to the fact that the root word of ψυχή is ψύχω, which means to breathe or blow cool air.
ψύχω(breathe):
ψύχω: 2 future passive, ψυγήσομαι (cf. Lob. ad Phryn., p. 318; Moeris, Piers. edition, p. 421, under the word); from Homer down; to breathe, blow, cool by blowing; passive, to be made or to grow cool or cold: tropically, of waning love, Matthew 24:12. (Thayer's Greek Lexicon)
Thus, Matthew 10:28 can be validly interpreted in this manner;
Matthew 10:28 "And do not fear those who bring to ruin the body but cannot bring to ruin the vital breath[as it is possible for God to resurrect you]. Rather fear him who can utterly destroy both the vital breath and the body in Gehinnom[thus rendering your entirety obliterated, with no way back, not even in a resurrection]."
The passage makes so much more sense now. How so? Well, because of Genesis 2:7;
Genesis 2:7 "...then the Lord God formed the man of dust from the ground and breathed into his nostrils the breath of life, and the man became a living soul."
Human beings consist of the physical(dust of the ground) and the spiritual(breath of life), which work together in harmony to construct a living soul(cf. 1 Corinthians 15:45). When we die, only the physical part of us is destroyed; the spiritual is not destroyed but returns from whence it came, i.e. God(cf. Ecclesiastes 12:7, Psalm 146:4). Thus, there is hope. God can resurrect us(bring us back from the dead) by incorporating our spirit/breath of life into a functional body to contain it, thus regenerating the harmony between the physical and the spiritual.
But how are you supposed to do that if the spirit/breath of life doesn't even exist? You can't put a nonexistent spirit into a body! If the spirit becomes nonexistent(via the only process that results in the spirit becoming nonexistent, i.e. destruction by God), then there is absolutely no way back; God can never resurrect you because He has chosen to permanently destroy the spiritual part of you. When God chooses to destroy, it is total destruction from which there is no return(that is why the more intense word, ἀπόλλυμι, is used to describe what God does to the body and spirit, as opposed to the less intense word, ἀποκτείνω). And this is why we must fear God; the ability to destroy the breath of life of a person belongs solely to God. No one else has it, thus we should not fear anyone else as though they are capable of destroying the spiritual part of us. Humans are capable of destroying only the physical part of us, our body; not the spiritual part of us, our spirit/breath of life. Only God has the ability to breathe the breath of life into us(Genesis 2:7), and in the same way, only He has the ability to destroy it.
Thus, the inner consciousness or mind(soul) is not in view in Matthew 10:28. Jesus is not implying that our conscious soul lives on after death; he is implying that our spirit/breath of life lives on after death. But that is someone we already knew from long ago!
Ecclesiastes 12:7 "And the dust returneth to the earth as it was, And the spirit returneth to God who gave it."
The point Jesus was trying to make at Matthew 10:28 is that we should not fear those who have no capacity to destroy our breath of life, but rather the One who alone has the capacity to do so, namely, God. That our inner consciousness(soul) remains after death is not something implied by the text whatsoever.
Hope this helps! Have a wonderful day. :)