Let's try and clear some things up both on a Scriptural plane and on a biological one that will make sense.
First, what does the Bible say:
Genesis 3:15 And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel.
--a prophetic reference to the seed of Mary.
Galatians 4:4 But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law,
--a direct reference to the virgin birth and the humanity of Christ.
John 7:42 Hath not the scripture said, That Christ cometh of the seed of David, and out of the town of Bethlehem, where David was?
--A reference to the human lineage of Christ back to David, a verse also using the word "seed" the Greek word "sperma."
Christ was fully human.
Christ was fully God.
He was the God-Man; wholly God and wholly man at the same time. The two natures are inseparable. He never at any time gave up his deity.
Now, as delically and tactfully as we can let's insert some biology here. What happens when a child is born?
The ovum of the mother is fertilized by the sperm.
Shortly after mitosis occurs, giving rise to two cells, which in turn give rise to four cells and continue to duplicate.
After about 3 days the embyo has become a mass of about 16 cells and makes its way to the uterus. There cell division continues.
What is important to note is that as the embryo grows it takes no blood from the mother. It produces its own blood, its own circulatory system, as it does all of its own organs. None of the mother's blood enters into the child.
Now, I will stop there. I hope that is enough biology. And I will return to the Scriptures.
Luke 1:34-35 Then said Mary unto the angel, How shall this be, seeing I know not a man?
35 And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.
Matthew 1:20 But while he thought on these things, behold, the angel of the Lord appeared unto him in a dream, saying, Joseph, thou son of David, fear not to take unto thee Mary thy wife: for that which is conceived in her is of the Holy Ghost.
These are the words of the angel, and also the Word of God--the explanation that we have of what happened.
In Luke 1:35 the angel refers to "that holy thing" in Mary, and rightly so. Mary did not understand biology. This was the beginning of the nine months. What was in Mary was just an embryo--a holy thing; a fertilized egg; perhaps a zygote. Thus the angel uses a term--holy thing, not contradicting any modern science.
Matthew makes it clear that she was conceived by the Holy Spirit, and not by man. Here is where you must break down and admit the miraculous of the virgin birth. The fertilized egg ("sperm" and ovum together) was only possible through the intervention of the Holy Spirit. It indeed was the "seed of the woman" made possible through the intervention of the Holy Spirit. That "holy thing" was made possible when the "power of the highest overshadowed Mary," just as the Scriptures said.
The sin nature is passed down through Adam (Rom.5:12). Thus it was necessary for Christ to be born of a virgin. Even so Christ had his own blood; not Mary's. He inherited some of Mary's characteristics, and even chromosomes. He had to. He had a human body; human flesh; just like you and me. But his blood was his own. That is a biological fact, as you can see.
Thus to give a clear answer--The blood that Jesus shed on the cross was not Mary's blood but His own blood.
He was still wholly man and wholly God at the same time.
I hope that clears up some questions, and is not too explicit at the same time.