The phenomenon regarding "gender identity" is real, but the underlying concept is incoherent, as I've said. This doesn't stop anyone from studying it, just in same way people study moral theories or theology.
what specifically is incoherent about it? I attempted to address that by noting that active areas of research will always have mutually incompatible theories that are incoherent when taken together, but perhaps I misunderstood what you meant.
You cannot "define what a woman is", but you can "identify as a woman" (what do you identify as?).
You cannot really say why being trans-gender is okay, but being "trans-abled" (so called amputee identity disorder) is not. Nor is being "trans-racial" (https://en.wikipedia.org/wiki/Hypatia_transracialism_controv...). Even though we have long believed that race is more socially constructed than gender.
The whole concept of "gender studies" as it is now is one of the least scientifically sound academic fields (https://en.wikipedia.org/wiki/Grievance_studies_affair). The field is full of people openly questioning the very idea of trying to be an impartial researcher in search of "truth" and instead treat it as an activist project (https://hipatiapress.com/hpjournals/index.php/generos/articl... - an example of a paper that in plain text advocates for using male students of feminist studies as "viruses" spreading the good news).
Most of the theories are unfalsifiable anyway - and are no more scientific that angelology or hamartiology.
Plenty of people think things are coherent that, upon conceptual analysis, collide into a reductio ad absurdum around every corner. There are honestly so many problems with it, but let's go off the top of my head:
- Self-identity: There is nothing special about sex/gender as a property that distinguishes it as a special case for self-id from other empirical properties of human beings, such as height, eye color, or hair follicle shape. Barring special criteria, if we allow self-id for one property, we must allow it everywhere. Everything goes.
- If one truly believes that gender is distinct from sex, then what motivates one to "identify" as something else?
- Many if not most transgender individuals emulate the opposite sex and their associated gender stereotypes. Surgically-implanted breasts and facial hair are secondary sex characteristics—their presence has nothing to do with gender. How those characteristics are presented or hidden are gendered expressions, but not their mere presence.
- Epistemically, we only know what it is like to be the sex we are. We can attempt to understand "what it's like" to be the opposite sex, but we can never know. Therefore, one can "identify as" something else, but that something else is just a facsimile.
