Not sure that’s fair, as it's maybe homophobia out of a twisted idea of "inclusivity" and "acceptance for everyone". Like they have taken their own progressive ideas to the extreme point that it turned back and undermined homosexuality.
It also supports gender non conformity and often seems to be a, "gender is a social construct", position.
Yes, you can also argue that it's connected to rigid gender roles, as they apparently think that if you like traditionally female/male things, then you may be the opposite sex, rather than just gender non conforming.
So you can argue for some sort of parallel with conservative thinking, but I doubt it's enough to fairly be saying, "you belong over there with the right-wingers".
And the right wing don't want them either.
Matt Walsh giving the other side of the argument:
https://twitter.com/MattWalshBlog/status/1683820604883869697
Feminism set the stage for trans activists by insisting for years that there are no significant or inherent differences between men and women apart from anatomy. They are the ones who came up with the idea that most differences between the sexes were “social constructs.”
Also, trans activist tactics of "no platform" and "punch a TERF" would appear to be borrowed from the left-wing "antifascist" movement.
And of course it was traditional lesbian and gay groups that developed into extending their support to trans activism.
If there was any "conservative thinking" involved, then I'm assuming it was just one thing in the mix with their progressive values driving it forward.