Great question. Well, it's really similar to being gay, lesbian, or bi.
The claim that trans women and men insist they are and always have been biological women or men is largely an Aunt Sally. Some trans people do say this, but in my experience most talk about it in a much more nuanced way, as people whose lives have involved a voyage towards becoming the gender they have always felt or known themselves to be.
This involves physical transformation to a greater or lesser degree because without it you can't experience the societal role or be accepted as the gender you are - including, but not restricted to, who you are when you have sex.
At the root of the current wave of transphobia is an uncritical acceptance of the gender-critical, biologically determinist position, which denies that the socialisation and lived experience of gender is as real as biological sex.
Gender IS real in the same way that orientation is. Just because it's experienced cognitively, emotionally and as a relation to your body, rather than a fact about your chromosomes and what they do, doesn't mean it's less real - because if you insist it is less real, you're saying only genes matter; cognition, emotion and endocrine make-up dont. Nature -defined very narrowly - triumphs.
The opposite to nature is not 'Nurture'; that implies people become trans because of traumatic or adverse experience and is, essentially, a pathology.
To say so opens the door to saying that's why people are gay, too. What lesbians and gay men have done is to successfully propose that it doesn't matter whether sexual orientation is innate or not, 'disordered' or not: I may have been 'born this way' to some extent, as a dreamy, slightly masculine kid, but that's not the issue. The issue is that I didn't decide to be a lesbian through confusion or adversity or rebelliousness, but because I experienced a largely somatic, hormonal rush of sexual longing when I, a girl, saw other girls and not when I saw boys. I then took a decision later on to socialise those feelings by adopting an identity called Lesbian.
I do not know what it's like to be trans; I've always been OK about inhabiting a female body. But I'm prepared to believe that for trans people, the clash between their body and their nature is as visceral, as immutable, and at times as profoundly ego-dystonic as I, growing up lesbian, found my sexual orientation to be.
The only difference between me and a trans person is that my orientation to others as sexual objects didn't fit society: their orientation towards themselves as sexual subjects doesn't.
But that's as far as I'll go: I don't presume to imagine what it's like being trans. Neither should you.
You see how far we've come from dying on the hill of 'men are men and women are women and nothing can change that'? As scientists often say, I think you'll find it's more complicated than that.... But you need to read beyond these echo chambers and be open to conflicting ideas and thoughts.
So yes it is very different to claiming to be black or disabled.
Hope that gives you something to challenge the orthodoxy of these threads. :)