There is no scientific or biological definition of gender. Because of that, there is no objectively correct answer to whether or not a tranny is their identified gender.
There's a deeper principle here at work: there's no objective definition of
any word. Words only have meaning in context.
Whether or not transgenderism is legitimate is a complex topic. I don't really much to say about that. I'm only arguing against people's attempts to claim that scientific definitions of words are
the definitions of words. They aren't.
People try to make the argument that a scientific usage of a word is the overriding one, because of the objectivity of science. It is true (as far as I'm concerned, anyway), that science is objective and the ultimate source of truth. But science doesn't say anything about how we should apply science outside of a scientific context.
All science has to say about these questions are that things like chromosomes exist and people have made statistical observations based on sexual characteristics like them. Certainly useful information, no doubt. But to claim that they actually fix what the English meaning of the words "man" or "woman" mean is a confirmation bias. It's not really supported by linguistics.
Science might as well be a separate language that you're merely translating into your native language. For example, there's the scientific definition of planet, compared to the astrological definition of a planet. Or "frog" and "toad" both have English definitions, but there's not really a consistent difference, scientifically.
Unfortunately, like most religious nutjobs, they keep trying to pretend their views are backed by science.
Well, I mean, there's certainly some interesting evidence to suggest that gender confusion might have some concrete roots.
But either way, I'm not really that concerned with the science, so much as the practical social understanding of gender.
If enough people accept a transman as a man, then the English language has evolved (like all languages have evolved to their current state) to accommodate that. Language is ultimately defined by usage.