I suspect that Mike Ashley, Mick Keane and Michael Gove share the same first name but they identify differently. I have the first name on my birth certificate and the completely different one by which I am known. I identify as male, but that doesn't mean I have to worry that male thieves make me a thief, or male Chelsea fans make me a Chelsea fan. As someone else has said, I can cope with people born female identifying as male. It is about feeling comfortable in knowing who I am.
The real issue here is surely the nurse's reaction. Whatever her specialism, it is almost inevitable that she will witness some challenging situations. If she is not able to react in a mature fashion to a disfiguring birthmark, a gruesome injury, a deformed physical characteristic, a person in significant psychiatric distress, what use is she to the medical profession and, more importantly, to those requiring her help?
Of course, none of us were there so we don't know if this was a genuine misunderstanding or an example of professional ineptitude. All we can say with certainty is that some people in this world are less accepting of others' individualities than they might be.