Or to put it another way, nobody says "I'm a transphysician. I feel like and identify with being a doctor. I want to wear a white coat and have people put "MD" after my name. This makes me a doctor, even though I don't know how to treat sick people."

But a lot of people want the status benefits of being a doctor without having to do the hard work of going throw med school or even knowing how to heal people. Come to think of it, this isn't a bad analogy of transwomen. A man want the status and attention that women get, especially in nerd or SJW circles. Furthermore, these days by declaring himself a "transwoman" he even becomes more special then ordinary women.

Sorry, the analogy isn't that close. I was just trying to get at the idea of there being a "doctor" node which is different from any one particular property of doctors, because it's a socially defined cluster, not a natural kind.

You're not a doctor if you don't treat sick people or have the ability to treat sick people.

Nurses also know how to treat sick people and treat sick people. That doesn't seem to be primarily what being a doctor is about.