This has probably been asked before, but I've only dipped into this megathread from time to time so I may have missed it.
I've heard enough from trans-activists to know that when someone identifies as a woman, for example, they do not have to adhere to any particular type of presentation, mode of dress, speech patterns, expectations of behavior, roles, life pathways, or anything else. They can pretty much do and act however they please while retaining their gender identity as a woman.
In this case, how is it any different from identifying as a "man", a "nonbinary person", a symbol (as the musician Prince did), or even a random collection of letters. In fact, it seems as the word "woman" when used as a gender identity has thereby just been reduced to a string of letters. It doesn't signify anything with any properties at all to distinguish it from "man", "nonbinary", or "xyzzy".
I think in linguistics speak, it's sort of like a signifier with no signified. Why does the gender identity term "woman" have anything whatsoever to do with the other word "woman" that is used to signify adult human females, other than that they share the same spelling?