(Please don't say XY chromosomes, you don't do a genetic test on everyone you meet.)
A her is someone who was born with a vagina.
The very, very few people who don't fit clearly into either of these categories (intersex) are exceptions and can be handled on a case-by-case basis.
They are immutable because their reproductive role is immutable. A her can never impregnate. A him can never gestate.
(Either can choose not to perform their particular ability, or lose the ability to do so, but that doesn't change the meaning of these words.)
If a car is broken down, is it no longer a car?
Does it become an airplane?
As for him gestating I encourage you to read more Sci-Fi.
I have a friend who was named at birth after their absentee parent and really dislikes ever being called by that name. It is important and polite to use names that people personally prefer within reason.
And if you really feel that your gender is being intruded upon by trans people then i encourage you to find something more meaningful and unique to attach your identity to, because your manhood feeling threatened is not a respectable reason to deny trans people their identity.
I was going to ask you the same!