I've known since I was a child that I didn't fit the usual picture of "male." I kept rejecting (in my mind) the whole bunch of ridiculous expectations put on people assigned male, and adjusted to them as little as I could get away with. As I grew older, I saw more and more of what girls and women face, and thought most of it was garbage, too. (I became a feminist pretty much as soon as I learned about it.)
Once I saw that I might be trans and that transition was possible for me, I started on that road. I think of myself as a refugee from "Manistan" fleeing to "Femininia." I now live as a woman, but only to the extent I feel like -- one thing feminism has done is widen the spectrum of what women are expected to be.
But I don't say that I have become a woman, any more than a refugee in the USA becomes a native-born (US) American. As for "identifying" as a woman -- I don't even understand what that would mean. I'm me, but being male or female (or non-binary) aren't essential parts of who I am. They're just how I have to relate to a world ruled by the gender binary. I fit myself into the female role because it fits better than the male role ever did, but it's not "who I am."
That's why I call myself non-binary. I've also noticed that I feel a lot more at home among non-binary people than among the binary trans folks -- it seems like a lot of binary trans folks are obsessed with stuff I don't care about.