You can't self-identify yourself as being of a certain profession because you first have to do it. I cannot declare "I am a nuclear scientist now, and you must address me as doctor", and then expect people to actually do it without questioning the veracity of the claim. Self-identifying as a species doesn't work because of DNA, self-ID as a sex doesn't work because of DNA, and self-ID of a profession doesn't work because it's based on knowledge, skill, and credentials (etc).
That's true, but there are identities where self-identity does make sense. Religious identity would be a good example. I think it's best to think of feelings-based gender identities as akin to religious identities. They don't replace the category of sex, but they are valid in their own right.
Sure, but that's not the only thing they're defined by. We might for example like to categorise people by their profession, and doing so is not in conflict with categorising them by their DNA.