When a preposition is part of the object of the verb, its meaning depends on its meaning in combination with the verb.
1) In general, words usually have multiple meanings based on the context. Here,
Compare the two phrases in v15 and v16 in John 3 (NA28).
ἵνα πᾶς ὁ πιστεύων ⸂ἐν αὐτῷ⸃ ⸆ ἔχῃ ζωὴν αἰώνιον. v15
ἵνα πᾶς ὁ πιστεύων εἰς αὐτὸν μὴ ἀπόληται ἀλλὰ ἔχῃ ζωὴν αἰώνιον. v16
These are parallel phrases giving ⸂ἐν and εἰς the same meaning as in these phrases. This doesn't mean the prepositions always have the same meaning. They do not. It means they have overlapping meanings where they can mean the same. The translations translate them the same. Wallace specifies that these two prepositions have overlapping meanings.
- confusion/overlapping of prepositions (e.g., εἰς/ἐν, ὑπέρ/περί).
Wallace, D. B. (1996). Greek Grammar beyond the Basics: An Exegetical Syntax of the New Testament (p. 20). Zondervan.
The lexicons give the variation in the means of words according to how they fit their context. BDAG suggests πᾶς ὁ πιστεύων εἰς αὐτὸν means "everyone whose faith depends on him."
ε. w. prepositional expressions: εἰς Ro 4:18, if εἰς τὸ γενέσθαι αὐτόν here is dependent on ἐπίστευσεν. πιστεύειν εἰς τὴν μαρτυρίαν believe in the witness
Arndt, W., Danker, F. W., Bauer, W., & and , F. W. (2000). In A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 816). University of Chicago Press.
BDAG indicates πᾶς ὁ πιστεύων ⸂ἐν αὐτῷ means "everyone with object of one's faith based on him." Thus, there is little difference in their meanings in these two phrases.
ἐν prep. w. dat.... ⑧ marker denoting the object to which someth. happens or in which someth. shows itself, or by which someth. is recognized, to, by, in connection with:...
J 3:15 (but ἐν αὐτῷ is oft. constr. w. πιστεύων, cp. v.l.);
Ibid. (3rd ed., p. 329)
2) While the New Testament was written in Greek, the New Testament writers were influenced by Aramaic and Hebrew.
While εἰς αὐτὸν in John 3:16 isn't a predicate nominative this suggests the use of εἰς can be influenced by the Semitic preposition ל when no preposition is necessary.
C. Substitution for Predicate Nominative (εἰς + accusative)
Εἰς + the accusative is occasionally found replacing the predicate nominative in the NT. Although this construction is found in the papyri, it is usually due to a Semitic influence (Hebrew ל). This idiom is frequent in OT quotations ...
Wallace, D. B. (1996). Greek Grammar beyond the Basics: An Exegetical Syntax of the New Testament (p. 47). Zondervan.
Note the hiphil of אמן in the Hebrew Old Testament. Thus, the Hebrew word for believe is usually followed by a preposition.
hif: pf. הֶאֱמִינ/מִינוּ, הֶאֱמַנְתִּי/תֶּם, impf. אַ/יַאֲמִין, יַאֲמֵנ/מֶן־ impv. הַאֲמִינוּ, pt. מַאֲמִין: causative —1. to believe = to think ... with לְ and inf., to be convinced that Ps 27:13; —2. to regard something as trustworthy, to believe in: a thing ... with בְּ, to (have) trust in ... with לְ ... —3. to have trust in, to believe in, God: with בְּ ...; with לְ ...; abs. to believe ...
Koehler, L., Baumgartner, W., Richardson, M. E. J., & Stamm, J. J. (1994–2000). In The Hebrew and Aramaic lexicon of the Old Testament (electronic ed., p. 64). E.J. Brill.
Thus, prepositions with πιστεύω may be an unnecessary addition from Hebrew influence.
Examples for preposition in Hebrew but not in the Septuagint (LXX):
καὶ ἐπίστευσεν Αβραμ τῷ θεῷ
οὖν μὴ πιστεύσωσίν μοι
אִם־לֹ֣א יַאֲמִ֣ינוּ לָ֔ךְ
ἐὰν δὲ μὴ πιστεύσωσίν σοι μηδὲ
וַיַּֽאֲמִ֨ינוּ֙ בַּֽיהוָ֔ה וּבְמֹשֶׁ֖ה עַבְדֹּֽו
καὶ ἐπίστευσαν τῷ θεῷ καὶ Μωυσῇ τῷ θεράποντι αὐτοῦ.
אֵֽינְכֶם֙ מַאֲמִינִ֔ם בַּיהוָ֖ה אֱלֹהֵיכֶֽם
οὐκ ἐνεπιστεύσατε κυρίῳ τῷ θεῷ ὑμῶν,
וְלֹ֤א הֶֽאֱמַנְתֶּם֙ לֹ֔ו
καὶ οὐκ ἐπιστεύσατε αὐτῷ
I'm not finding examples in the LXX of πιστεύω with ⸂ἐν or εἰς preceding the object, but lots of examples of πιστεύω with the object dative and no preposition. In Hebrew the hiphil of אמן, when it has an object, the object is preceded by ב or ל.
While the LXX was Koine Greek, Alexandra was known to have more of a classical Greek grammar. In textual criticism of the New Testament Alexandrian texts were known for making corrections more in line with classical Greek grammar.
Septuagint. The Greek translation of the Hebrew Bible. This Greek translation was undertaken by the Greek-speaking Jews in Alexandria from the third to the second century B.C.
Patzia, A. G., & Petrotta, A. J. (2002). Pocket dictionary of biblical studies (p. 105). InterVarsity Press.
While the LXX is known for semitic influence on the language, when it came to πιστεύω, the dative object with no preposition followed more of a Greek rather than a semitic style. Nevertheless, it shows that the prepositions ⸂ἐν or εἰς with the object are unnecessary.
Both New Testament Greek and Septuagintal Greek are considered substrata of the Koine. (The LXX, however, is so heavily Semitized-precisely because it is entirely translation Greek-that it is normally treated as in a class by itself.)
Wallace, D. B. (1996). Greek Grammar beyond the Basics: An Exegetical Syntax of the New Testament (p. 17). Zondervan.
Appendex of references
§ 7. “Believing,” in the Fourth Gospel
 It remains to consider the Johannine traditions about “believing,” or “trusting.” The best way of doing this will be to note the different expressions, (“trust (absol.),” “trust (dat.),” “trust to (εἰς),” “trust to (εἰς) the name of,” “trust that,”) in the order in which the Evangelist introduces them, and to trace their principal recurrences, so as to give an outline of his doctrine as expressed in Christ’s words and in Evangelistic comments. Here it may be observed that “trust in” and “trust on” are not mentioned. The former, since it occurs only once in N.T., might well not be used by John: and indeed “abide in,” rather than “believe in,” represents his doctrine about the highest and ultimate relation of the believer to God. “Trust on” also, would be inconsistent with his view, which is, that man does not “rest on” Jehovah as on the Rock of the Psalmist, but that he is “in” the Father—as a child is “in” his father’s house, or “in” his father’s heart.
Abbott, E. A. (1905). Johannine Vocabulary: a comparison of the words of the fourth gospel with those of the three (pp. 32–33). Adam and Charles Black.
 3:16–18 “For God so loved the world that he gave his only begotten Son that everyone that trusteth to him might not perish but might have eternal life … He that trusteth to him is not under judgment (οὐ κρίνεται). He that trusteth not is already judged [guilty] because he hath not trusted to the name of the only begotten Son of God.” The comment of Barnabas on the healing efficacy of the Serpent may be of use here: “When any of you shall be bitten (saith the Scripture) let him come to the Serpent that is hanging on the tree and let him hope and believe that it, though dead, is able to make alive and straightway he shall be saved (i.e. healed).” This is a very rudimentary and erroneous definition of “trusting”: but it helps us to understand why John does not attempt to define, and prefers to suggest. And his suggestion here is that we are to trust—not in a “dead” person or “thing,” nor that a person or thing can “make alive,” but—to (εἰς) an “only begotten Son,” who will make us alive (as will be shewn hereafter) not in spite of the fact that He has died, but because He has died (as the seed dies to live and to give life).
-- (Ibid. p. 44)
2. πιστεύω εἰς and πιστεύω with the Dative.
Specifically Johannine is the fact that πιστεύειν with the dat. can be used for πιστεύειν εἰς (→ n. 221); the linguistic variation contains no material distinction. Thus in Jn. “to believe Jesus when He preaches (or tells the truth, 8:40, 45), or to believe His Word (2:22) or words” (5:47), is equivalent to “believing in the Jesus who is proclaimed.” This corresponds to the fact that Jn. achieves a unity of Proclaimer and Proclaimed not yet attained in the Synoptic presentation. In this respect Jn. is not correcting the Synoptic depiction. One might rather say that he is correcting the kerygma. He wants to make it plain that it is the One proclaimed who Himself meets and speaks with us in the kerygma. What the kerygma proclaims as an event, God’s act, has itself the character of word. For this reason Jn. can call Jesus Himself the Logos (1:1). In this way he radically develops the thought that God’s word and act are a unity. In the word we meet God’s act, and in God’s act is His word, → 215, 29. ἀκούειν can be equivalent in meaning to πιστεύειν. “To believe in Him” is “to come to Him,”346 “to receive Him” (1:12; 5:43), “to love Him.”
Bultmann, R. (1964–). πιστεύω, πίστις, πιστός, πιστόω, ἄπιστος, ἀπιστέω, ἀπιστία, ὀλιγόπιστος, ὀλιγοπιστία. In G. Kittel, G. W. Bromiley, & G. Friedrich (Eds.), Theological dictionary of the New Testament (electronic ed., Vol. 6, pp. 222–223). Eerdmans. [While I don't agree with Rudolf Bultmann's theology, he was a good linguist.]