In the case of transgender people, I'm skeptical that the gender identity that is such a cornerstone of it is a truly innate element, or whether it's a reflection of gender being a social construct, and being taken as an identity.
As a caveat, the trans* umbrella has grown fairly large over the last decade or so, so I can't rule out that there are those who gender expression is based far more on social constructs than an intrinsic issue with the relationship between mind and body. When I say 'mind', I'm using that term rather loosely. I am talking about neurological processes one way or the other rather than some abstract philosophical concept. Whether that is due to have a brain which is at least partially feminized/ masculinized, or it's due to some other phenomenon, we're talking about something occurring in the brain.
My dysphoria is a result of mismatch between mind and body. For whatever reason, they don't align, and I have been aware of this from a very very very young age. The idea that there should be some sort of mental mapping to sexual characteristics may seem alien to someone who has never experience this incongruity, but that doesn't necessarily mean it isn't normally the case for people regardless of gender identity. As a vaguely akin (though not completely analogous) concept, when you open and close your hand, does that feel like a process which takes place wholly in your hand and arm, or do you feel your brain making it happen? Ordinarily, we'd just move our hand without any pressing awareness that our brain was active in that process as well. But perhaps if you suffered from certain types of apraxia, or you were deprived of a limb and experienced phantom pains the broken link would make you far more cognizant of how the brain and hand were mapped together.
Like I said, it's not truly analogous. The point is merely when things are in alignment and working, it's easy to not notice any distinction between mind and body. This idea that one's mind could expect to find different genitals than what are actually there may seem bizarre to someone who doesn't experience it because they ordinarily would feel any distinction between a neurological sense that they should have a penis with the fact that they do have a penis. It would all just feel like having a penis. When mind and body don't align, that ends up being the root of dysphoria. And what is felt in the mind is the seed for identity.
Even if the mind/ body disconnect does form the root of dysphoria, this doesn't mean social constructs are irrelevant. We are social creatures (even the fiercely introverted such as myself). We understand how people are sorted and how pervasive engendered concepts really are. As a transgender woman, when I was a child I understood what my mind told me and how it did not align with not only my body, but with how i was seen and sorted by society. Every instance where I was excluded from things deemed female and every instance where I was included in things deemed male, it called attention to my dysphoria. It also placed distance between me and others because it was like there was this unseeable aspect of my mind which was entirely obscured by my genitals. How that all plays out between a bizarre property of my mind which doesn't properly align with my physiology, and how that all manifests into social constructs is a really complicated thing. Those social constructs do end up have impact on my identity which in turn feeds back into social constructs. But the root of it was less to do with with social constructs. As long as my mind and body couldn't be reconciled adequately, the dysphoria persisted.
Experiences vary amongst transgender people for a great many reasons. Still, it feels like a lot of dialogue is broken because we need to talk in terms of either biology or social constructs. We need to talk in terms of what is innate or learned/ conditioned. But there easily exists a reality where both are relevant not just for transgender people, but for nearly all people. It's just, transgender people have much more of an aggravating factor calling our attention our gender identity that most people in general.