@OvaHere
One of my sons is on the spectrum (he's diagnosed). He's 16 and so far has shown no interest in gender identity politics to my knowledge. I can see where it would spring from though.
He has in the past displayed quite rigid views about gender roles which in my opinion stems from a desire to categorise everything, which I think is one way he tries to make sense of the world. It's not really that different to what we all do to some extent he's just a bit more black and white about it than the average person without an ASC.
For example a couple of years ago his younger brother had his ear pierced and chose one of those fashionable diamanté style earrings. He couldn't stop going on at him and us that this was a 'girls earring' and his brother shouldn't have picked it because he is a boy. He was quite fixated on it for a while.
I had to spend a lot of time reiterating that it didn't matter, that 'things' don't actually belong to boys or girls and all the other debunking of stereotypes. He eventually let it drop but he does have a tendency to get a bit of a fixation about these sort of things. I can completely see how he would be susceptible to the sorts of explanations groups like Mermaids churn out.
I'm convinced the autistic desire/tendency to categorise things to understand them is part of that, I remember when I was about your son's age, I would draw Venn diagrams and write set theory about which sorts of social interactions were acceptable with which sorts of people - eg you can talk about personal issues with family and close friends and your medical professionals, and you can hug family and friends, and of those you can only do interactions that involve touch if you ask first (either vocally ask or imply an ask, say, by holding your hand out and they can decide to shake it, or open your arms wide and look at then and they'll decide to hug or decline it, that sort of thing). Whenever I see these lists on places like Tumblr or Reddit of endless "genders" and "sexualities" other than hetero/homo/bi, like "moongender" or "neutrois", I think, "classic autistic behaviour!" We like labels and definitions and clarity, and although the whole gender thing looks really opaque to me, it must have some meaning or explanation to these kids. On the one hand I identify with their need to categorise and classify everything, but on the other it scares me how easily I could've fallen for it, and I wish they had support to understand the world without this gender lens on it. They could learn about different personalities or different social interactions and then learn that each person has their own likes and dislikes and behaviours, and as they get to know more people they'll get a better idea of the variety and how these categories are more of a guideline than a hard rule, and a lot of "gender" rules are rubbish and extremely narrowly culture specific anyway.