I care a little bit. I don't have a strong opinion on the labels people give themselves. But being forced or socially pressured to speak untruths, or to go along with others speaking untruths, is very 1984ish. Altering language to obfuscate the truth is doubleplus ungood. And at its core, the left's demand that I say and believe a M-to-F transgender person is a woman is simply that.
I'm willing to work with them on the limits of language, so if that M-to-F transgender person wants me to use she/her pronouns, I'm happy to extend the courtesy. It doesn't seem quite right, but I don't have a better solution that's usable and respectful of their choice. I have a little less patience for made-up pronouns like "ze" or using "they" to address a singular person - now we're voyaging into the uncanny valley of grammar, and I can't even tolerate people who write **** like "where r u? c u tonite" in texts.
And I say this as someone who's on the extreme end of the personal liberty wing of the libertarian party. I think anybody should be free to do anything they want so long as they don't harm another person, whether it's cooking meth while smoking crack, riding a donorcycle without a helmet, having sex with whatever willing partner(s) for fun or profit, aborting 10wk fetuses for whatever reason, owning semiautomatic rifles even if they're black and have shoulder thingies that go up, altering their bodies with tattoos or hormones or surgery, living in New York, even listening to jazz if that kind of obscene auditory assault is their perverted idea of fun. Demanding that others lie to help you suspend your own disbelief about what you think you are or wish you were, is a big ask.