I think each person's identity, like their history, is unique and "over-determined." Trying to tease out what part is innate and what comes from experience for any particular person is likely to be impossible.
In my case, I don't recall having "transgender" feelings as such as a child. I do remember that I was unable to act the way a boy was supposed to, and I was already getting harrassed for it by the time I was 8 (I don't remember much before age 10.) And I remember that the idea of a boy getting turned into a girl was terrifying when I first heard of it (Tip -> Ozma in The Marvelous Land of Oz) I've also been aware of gender expectations as far back as I can remember, and I can remember finding them just plain stupid and weird. I do remember as a teenager finding a dress of my mother's which she no longer wore and wearing it while hiding in the cupboards under the eaves in my bedroom. Wearing "women's" clothes was both scary and attractive.
So maybe I can say I had "crypto-transgender" feelings.
On top of that I have C-PTSD, probably due to feeling emotionally abandoned at a very early age (possibly as early as 18 months.) I do remember that I had all kinds of trouble dealing with school, with other children, with adults, with expectations, and my parents were not only no help, but frequently did the sorts of things that got the message across that asking for help would only ever make things worse for me. To the end of their lives, they kept doing things to give the impression that they'd rather not be reminded that I existed. And no one else showed any willingness to understand or support me. To this day I struggle with feelings of worthlessness and wishing I were dead. On good days I don't notice those feelings, but on bad ones, all I can do is hold myself and bite my fingers and try in vain to cry. I have no idea how much this affected my gender identity (or non-identity.)
I often wonder whether, if I'd grown up in a different world and a different family, I would have been okay with being male. If being gentle and affectionate, not being competitive or domineering, hating fighting of any kind, being not especially coordinated (though I've read that trauma can cause lack of coordination), and kind of nerdy had been accepted as a perfectly normal and appropriate way for a boy to be, maybe I would have been happy with my body and my assigned gender.
But I didn't grow up in that family or that world. I am what my history has made me, and even if by some miracle the world would change and what's left of my family would change, I don't think I would be able to feel comfortable as a man, any more than someone whose leg has been blown off in a war will suddenly find it growing back once the war is over.