The term for "rib" in Gn. 2.21 is not an anatomical term, but more of an architectural one (like the wing of a building or the side of a building or room). The ancients knew nothing about surgery; they are not claiming that God sedated Adam and removed a rib to generate Eve. Instead, what the text is saying is that Eve is made of the same "stuff" as Adam (bone of his bone and flesh of his flesh). She is made of the same basic materials as he is; she is a genetic and biological human being as he is (though they knew nothing about biology or genetics. In other words, she is completely and distinctly human, as he is. She is his counter-partner, his equal. The emphasis is on their relatedness and the unity of humanity.

So we are not to think that God is taking a chunk of Adam and going, "Oh, great. Now what do I do? This is XY and I need XX. Drats." Instead, God is making them male and female (Gn. 1.26). We can also conclude that Eve is genetically different from Adam. They're not like identical twins, but one female and one male. Instead, they are separate, unique beings with separate and unique genetic codes, but she is not inferior. Instead, she is just as much in the image of God (1.26), just as much human (2.23), and just as much his equal (2.18).

Is Eve genetically identical to Adam except that instead of his Y chromosome, she has 2 identical copies of his X chromosome?

It's said she was made from his rib so there would only be his genetic information there. There also is the phenomenon of uniparental disomy monosomy rescue where if an egg or sperm is missing a chromosome, sometimes the baby soon after conception duplicates the chromosome it only got 1 copy of so it can then develop normally.