"... a person is more than their body ..."
That seems clearly true. It's wholly compatible with the fact that people are essentially embodied, though, isn't it? What do you think?
(You need to watch out for the Straw Man here. It's quite common: "No woman has a penis." ... "Aargh! Reducing people to their genitals!" ... Erh, no. --Why? Think about it.)
As for gender identity not being born in a vacuum. Very likely so. It doesn't follow the notion has any sense or coherence, though, does it?
My friend Dan's proof of the squaring of the circle wasn't born in a vacuum. And it draws inspiration from pre-existing ideas that persist on many sides of lots of conversations. Nevertheless we know it to be fallacious. There is no such thing as "squaring the circle" (in the sense Dan intends); nor could there be.
Likewise, there is no such thing as "gender identity" (in the sense certain trans apologists intend); nor could there be.
Do you think differently? What do you think "gender identity" means, if so?