The Greek text of Phil. 4:3 according to the Textus Receptus states,

Γʹ καὶ ἐρωτῶ καὶ σέ σύζυγε γνήσιε συλλαμβάνου αὐταῖς αἵτινες ἐν τῷ εὐαγγελίῳ συνήθλησάν μοι μετὰ καὶ Κλήμεντος καὶ τῶν λοιπῶν συνεργῶν μου ὧν τὰ ὀνόματα ἐν βίβλῳ ζωῆς TR, 1550
The Greek word σύζυγε is declined in vocative case, either the feminine or masculine gender, and singular number, from the lemma σύζυγος, an adjective functioning substantively in this particular context.
Regarding the declension σύζυγε, Lexigram states,

According to BDAG (which specifically addresses the question at hand),1

According to LSJ,2

According to Thayer,3

The adjective σύζυγος is related to the verb συζεύγνυμι, which means “to be yoked together.” This verb occurs twice in the Greek New Testament, and in both instances, it refers to being married.
For example, in Matt. 19:6, it is written,
6 so that they are no longer two, but one flesh. Therefore, what God has yoked together, let no man separate!
Ϛʹ ὥστε οὐκέτι εἰσὶν δύο ἀλλὰ σὰρξ μία ὃ οὖν ὁ θεὸς συνέζευξεν ἄνθρωπος μὴ χωριζέτω TR, 1550
The Lord Jesus Christ is referring back to the marriage of Adam and Eve.4 Accordingly, one who is σύζυγος, “yoked together,” would be one’s spouse. On the other hand, as the lexicons clearly state, the word does not only possess the meaning of “spouse.” Furthermore, the apostle Paul clearly implies in his first epistle to the Corinthians that he was unmarried.5
As for the claim that the apostle Paul is referring to someone’s name, I hesitate to accept this claim since he modifies σύζυγε by the adjective γνήσιε, which he does nowhere else before a name.6 On the other hand, he does modify names by other adjectives elsewhere,7 which therefore leaves the possibility.
By itself, it cannot be determined whether σύζυγε is referring to a male or female, since it could be declined in the masculine or feminine gender. However, because it is modified by the adjective γνήσιε, which is certainly declined in the masculine gender alone,8 then σύζυγε must also be declined in the masculine gender in agreement with the adjective γνήσιε.
In his commentary on Phil. 4:13, Heinrich August Wilhelm Meyer wrote,9

If the apostle Paul would have modified σύζυγε by γνησία, which is declined in the feminine gender (whether vocative or nominative), then there would have been a basis for the assertion that σύζυγε is referring to a wife.
Contrary to my assertion that σύζυγε is not referring to a name, Meyer also writes,10

Summary
In summary, σύζυγε by itself can mean “wife,” or even “husband.” Only an associated adjective or the definite article when the word is not declined in the vocative case could clue the reader in on whether it’s referring to a male or female! However, because the apostle Paul modifies σύζυγε by the masculine-gender adjective γνήσιε, the word σύζυγε in Phil. 4:13 is understood to be declined in the masculine gender and could not therefore be interpreted as referring to the apostle Paul’s wife. It could however be a proper name, but ultimately, the matter remains unsettled.
References
Arndt, William; Bauer, Walter; Danker, Frederick William. A Greek-English Lexicon of the New Testament and Other Early Christian Literature. 3rd ed. Chicago: U of Chicago P, 2000.
Liddell, Henry George; Scott, Robert; et al. A Greek-English Lexicon. 9th ed. Oxford: Clarendon, 1940.
Meyer, Heinrich August Wilhelm. Critical and Exegetical Handbook to the Epistles to the Philippians and Colossians, and to Philemon. Trans. Moore, John C. Ed. Dickson, William P. New York: Funk, 1889.
Wilke, Christian Gottlob. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. Trans. Thayer, Joseph Henry. Ed. Grimm, Carl Ludwig Wilibald. Rev. ed. New York: American Book, 1889.
Footnotes
1 p. 954
2 p. 1670
3 p. 594
4 Gen. 2:24
5 1 Cor. 7:7–8
6 cp. 1 Tim. 1:2; Tit. 1:4
7 e.g., “the beloved Persida” (Περσίδα τὴν ἀγαπητήν; cp. Rom. 16:12); “the beloved Apphia” (Ἀπφίᾳ τῇ ἀγαπητῇ; Phlm. 1:2); etc.
8 The corresponding feminine declension would be γνησία.
9 p. 161
10 p. 162