sex is not an immutable characteristic. that's where people go wrong with this - they don't understand the science properly. what they do is they get this religious idea that we're assigned a gender at birth mixed up with the science - which says no such thing - and then deduce it must be science because it's "common sense".
so, what does the science say, here?
i'd imagine most people will tell you that you're born with xx or xy chromosomes and that determines your biological sex, but that is actually biologically, scientifically incorrect. the chromosome is actually just a bit that codes for the hormone, and this is repeatably demonstrable and very well understood. so, if you take an xx embryo and flush it with testosterone, you get a biological male; conversely, if you take an xy embryo and flush it with estrogen, you get a biological female. so, let's put the question of chromosome editing off for a minute, because it's secondary to the point - biological sex is not defined by chromosomes, it's defined by hormones. and, from time to time, errors occur and you get exactly this outcome.
it follows that the definition of a biological male is an individual that is testosterone dominant, rather than an individual that is born as xy; likewise, it follows that a biological female is estrogen-dominant, and not somebody born as xx.
and, once you get the definitions correct, it is easy to see that hormone therapy can, indeed, change a person's sex - which is a hormonal, chemical thing and not something embedded into the dna.
now, you can argue that while it may be true that embryos are not sex-specific, the introduction of one hormone or the other to the embryo produces an irreversible process. but, that position is incoherent in the fact of a set of surgeries that can, in fact, reverse it. i mean, if you want to be empirical, there's your evidence - the surgery that replaces one physical vestige for another.
but, it's missing the point in the broader context, about physical sex organs being consequence of the hormones and not definitions of biological sex, itself. if you have this flap of skin or that one, it's still just some flaps of skin - it says nothing of the chemistry defining the concept of sex.
to my knowledge, we can't yet exchange testicles with ovaries or vice versa, but some species can, indicating how closely these structures are truly related. and, it's really just a matter of time. and, then what does empirical evidence state, clearly, in front of your face?
conversely, if there is a biological process to change one's skin colour, i'm not aware of it. but, i mean, it's a triviality regardless - and not defined by functional hormonal differences that alter how reality is experienced. sex is at least chemically, biologically real - race isn't.