What I'm trying, clumsily, to ask is, do you think the desire to present and be read socially as belonging to your sex class, and distinct from the opposite sex class, through the use of whatever cultural gender markers exist at the time, is what TRAs mean when they talk about "gender identity"? And if so, do you think that maybe it actually is innate?
Speaking largely about my intuitions and personal thoughts, I do think there is such a thing as gender or sexual identity. There are all kinds of identities, we seem to be made in such a way that we build them up from our experiences and cultural environment. From what I understand there is a lot of psychological study related to how we build and interact with identity, that has nothing to do with our sexed bodies, how you conceptualise yourself as a person, as a member of a family or race, and so on. So I would say our identification with our sex is one facet of that, probably a pretty basic one since sex, unlike race or favourite band, is ubiquitous and important for species survival.
I don't think this is something we can avoid, it seems to be integral in some way to our consciousness and I would say also having regulated personalities, and on a larger scale to social stability. I think its a big part in creating societies that function, and maybe the more complex the society is, the more it has to build up complex identities. (And maybe more can go wrong?)
It does seem to me that at least some of the classic gender dysphoria descriptions could be about some kind of fault in the way the individual forms their identity.
In the older cases it was associated by the doctors with homosexuality, and maybe that played a part in some way - perhaps it is even significant that we recognise among homosexual persons definite types (not sure what other word to use there, but like butch lesbians or effeminate gay men) that seem to take on some characteristics of the opposite sex. Those people seem to have a stable identity as male or female though, though many do have a period where they really struggle even more than heterosexual peers. And it also makes sense to me that the ones who can't make a stable sense of identity might have other related issues.
So my thought is that its a disorder liked to particular ways identity forms. I don't think all TRA are thinking about that though, the queer ideology has changed things a lot.