The Bible reveals that they sin nature is transferred biologically to the flesh in reproduction by the male's sperm.
That is why it says sin entered the world through one man (Romans 5:12), even though the woman was the first sinner.
The woman's ovum does not transfer the sin nature.
That is why we see the odd wording found in Genesis 3:15.
And I will put enmity
Between you and the woman,
And between your seed and her Seed;
He shall bruise your head,
And you shall bruise His heel.”
There was no mention of the man's seed in Genesis 3:15.
The woman's ovum itself does not introduce the defective genes to produce a sin nature.
If Adam never fell?
His seed and Eve's ovum would not have had the sin nature enter the world!
For their child would have been born without a sin nature, like Jesus was.
That is why God used Mary's ovum to produce the body of Jesus.
The Holy Spirit provided the perfect genetic materials needed to fertilize the ovum of Mary!
I believe mankind is called Satan's seed because Satan introduced the fall of Adam which resulted with man having a fallen nature.
Like, George Washington is referred to as the father of the USA.