The answer to your question is relatively simple. Mary provided the egg; God provided the sperm. This may sound shocking, and there is no single verse which "proves" my assertion, but there are many scriptures--particularly Romans chapter 5--which clearly lay out for us the reason why Christ was conceived and born with a sinless human nature. (The "God" part of His being is the reason why. Simply put, God cannot sin, nor can He even be tempted to sin. James 1:13)
Why do I link a "sinless nature" with the DNA of both the God-created part and the Mary-donated part? Let's look in detail at Romans 5:
". . . through one man sin entered into the world, and death through sin . . . [D]eath reigned from Adam to Moses, even over those who had not sinned in the likeness of Adam's offense, who is a type of Him who was to come [viz., Jesus]. . .. [If B]y the transgression of the one, death reigned through the one, much more those who receive the abundance of grace and of the gift of righteousness will reign in life through the One, Jesus Christ. So then as through one transgression there resulted justification of life to all men. For as through the one man's disobedience the many were made sinners, even so through the obedience of the One the many will be made righteous" (vv.12-19, excerpts NASB Updated).
The Argument and Proof of Divine Conception
Notice how Paul lays out his "case" for how the justification provided by God through Christ works in a "legal" sense. Through the use of rhetorical techniques and simple logic Paul lays the foundation for his assertions in Romans 5. Notice in the above verses and elsewhere in the chapter Paul repeats the words much more (vv.9, 10, 15, 17). These words indicate his use of the a fortiori argument, which has been defined as follows:
"An a fortiori argument /ˈɑː fɔːrtɪˈoʊriː/1 is an "argument from a yet stronger reason." For example, if it has been established that a person is dead (the stronger reason), then one can with equal or greater certainty argue that the person is not breathing. "Being dead" trumps other arguments that might be made to show that the person is not breathing, such as for instance, not seeing any sign of breathing.
"An a fortiori argument draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in the first. The second proposition may be considered "weaker," and therefore the arguer adduces a "stronger" proposition to support it" [my emphasis].
The proposition of which Paul speaks is that of representation. Adam represents one thing; Christ represents another thing. Whereas Adam represents humankind in general, Christ represents God. Now if the weaker proposition is true (viz., death reigned in humanity from Adam, onward, because of his one sin), then the stronger proposition ("the gift of righteousness will reign in life through the One, Jesus Christ," v.17) is even truer.
Paul, then, as a master lawyer arguing his case, wraps up this section of his forensic oratory in vv.18-21 with several "so" statements. In other words (using a little sanctified imagination):
"So we can see, your honor and honorable members of the jury, that I have proved this part of my case beyond a reasonable doubt."
Representational theology is at the heart of the answer to your DNA question.
Adam represents humanity in his disobedience, sin, and spiritual and physical death.
Jesus Christ represents humanity in His obedience, sinlessness, and spiritual and physical life.
Adam's "contribution," his "gift" (notice the implied sarcasm in Paul's use of the word) to humanity is sin and death. (Thanks a lot, Adam!). Literally, then, Adam's DNA, as it were, is the agency of successive, generational sin up the to present day and beyond. That "Y chromosome" you refer to is tainted, as it were, by sin. Is this literally so? I do not claim to have an answer to that question. Theologically, however, it is so.
Jesus' contribution, His gift to humanity is righteousness, life, and justification. Literally, then, Jesus' DNA, which originated in God, has the power to pass on to all those who believe in and receive Him, those three gifts, and more!
The "sperm" donated by a holy God to the process of fertilization and conception, and the miraculous creation of a sinless embryo, was impeccable--without sin, sinless--in contrast to a mere man's contribution to fertilization and conception, which is peccable. King David put it this way:
"Behold, I was brought forth in iniquity,
And in sin my mother conceived me" (Psalm 51:5)
Then prophetically--and certainly unbeknownst to him, speaking of his Savior to come he said,
"Behold, You desire truth
in the innermost being
And in the hidden part
You will make me know wisdom" (v.6).
Christ, the Believers' Representative
Whereas David was speaking on one level in this verse about God's desire for all of us to be truthful with Him whenever we sin, and not to attempt to cover up our sin or pretend it does not exist (David, you will recall, committed adultery and murder-by-proxy), David was also speaking on another level about the One who was to come, the One who would one day say,
"I am the truth" (John 14:6)
Moreover, in His relation to all believers in Him--
"to those who are the called, both Jews and Greeks, Christ [has become] the power of God and the wisdom of God" (1 Corinthians 1:24 NAS). Many evangelical Protestant churches affirm not only the truth, who is Christ, but they often quote John 1:14, which is source of comfort to all Christians:
"And the Word became flesh and dwelt among us, and we saw His glory, glory as of the only begotten from the Father, full of grace and truth" (my emphasis)
I highlight the word grace because Jesus, in a sense, did not lead with truth, but grace. The truth is, we're all messed up because of sin. The good news is, He is gracious in looking beyond our fault and seeing our need. Ah, the simplicity and power of the gospel!
In conclusion, the theological concept of representation is a powerful argument which really provides a satisfying answer, I believe, to the heart of your question. Christ's DNA was impeccable. Would it look the same under a microscope as the DNA of any other representative of the human race. Yes. Was it the same DNA? No. It was the DNA of the God-Man, whose life, death, and resurrection became the basis for our justification before a holy God.