It’s become a really strange discussion.
I think that at least part of the problem stems from a kind of variant of the naturalistic fallacy. People feel that if they can say that they are the way they are, or behave the way they do, because of nature, then this means that any social/moral judgment which disapproves of them is negated. And so there is some impulse, when people are challenged over their gender expression, to say something along the lines that this is an immutable and real part of my nature, ergo, not open to criticism.
This, however, comes into tension with the fact that many aspects of “gender” are culturally defined. I can be a natural biological male, by virtue of genes and development. But I cannot be a natural middle-class American guy, because most of what goes into that is not biologically determined. That cannot, really, be said to be innately who I “am” but it is certainly the mode of my gender expression. This stuff is always socially/culturally mediated.
And so the very people who insist that sex and gender are not the same thing often blend the two when trying to validate their transgendered status against social objections: they point to nature to say, well, sex is complicated. But transgenderism isn’t about complicated sex; it’s about complicated gender.
I’m not much for the naturalistic fallacy. I think that socially we do best by accommodating and acknowledging how others want to express themselves, gender expression being just one of many aspects of that.
One way I have framed this in my own mind is in relation to the many arguments people have had about whether homosexuality is innate or “chosen.” The merits of that dispute aside, I always think: why should that matter? Hypothetically, if homosexuality were (to be clear, I do not think it is) 100% “chosen” and not at all innate, the silent implication in those arguments always seems to be that, well, then it’s all right for us to declare it to be evil and to treat homosexuals badly. But that implication is, I think, just wrong: it wouldn’t really matter because, as @T_aquaticus points out, “it’s none of my business.” Our moral responsibility to others includes accepting differences, however those differences arise, when those differences do us no harm.
P.S. On a minor note, I find the notion of sex being “assigned” at birth, as people say these days, a bit strange. It seems to me that it is “inferred” at birth from the then-available evidence. “Assigned” makes it sound like somebody made a discretionary decision. Now, the inference can be wrong, of course – Androgen Insensitivity Syndrome or other developmental anomalies can complicate the matter.