I have no problem with how people want to identify themselves. The issue is that they expect everyone else to address them in the way they want, depending on what colour their toenails are, what season it is or what day of the week it is, and get all offended If someone gets it wrong. That's the issue.
People can identify as a fudging squirrel for all I care, but I reserve the right to address them as a person of one of two genders we have.
Who exactly is the "they" you're referring to? With the toenail / day of the week thing, I'm not sure it helps to comedically exaggerate something you might not understand in an attempt to make it appear preposterous.
I'm not certain which right you have to address them as a person of "one of two genders", it seems like you have gender mixed up with sex. It shouldn't really come to rights / the legality of anything anyways. My ex from uni days now identifies as non binary, should it be my right to refer to them as her? What would I gain from doing so other than not respecting someone's request? Genuine question, it doesn't feel like a whole lot is being asked if me so what's the problem?
Without having to get in to all of it, it's simple enough to just learn someone's name so then use of personal pronouns shouldn't even have to come up.
There's people starving, bombs flying, wars going on, a lot worse stuff than simply being asked to refer to someone you assumed was a he or a she as something other than your initial assumption. A lack of kindness / empathy / compassion and a strongly fixed belief system which ignores other cultures that have existed for centuries with an appreciation of different gender identities seems to be "the issue" more than what you've stated.
It seems the people most worked up by this are folks that have had no real life experience of talking to trans / non binary people and are just wanting to moan. If you had a son / daughter who genuinely felt they were born in the wrong body and transitioned, would you refuse them the courtesy of addressing them how they felt comfortable? If you aren't comfortable answering that question or even pondering it then fair enough, that could be quite personal but it's just a thought exercise, no offense / judgement intended.
Fun fact is that all embryos began life as female so all males are effectively trans when you think about it, and some folks are born with both male and female bits and bobs, the parents have to choose but their choice doesn't always align with how the child will identify when they grow up. It's not a simple subject that's for sure!