Yet entirely objective well over 99.5% of the time. Even most ranspeople can be clocked as the gender they identify with, if not flawlessly passing. An accuracy rate that high rises to objectivity in the practical sense.
"Hey, that dude is in a dress and lipstick, I bet he identifies as a chick" isn't a hard leap to make. What's more challenging is determining whether a male in a pink t-shirt with a unicorn on it, jeans, and sparkly chucks identifies as a female or whether they're simply into sparkles. Or determining whether the female in a metallica shirt, jeans, and steel-toed boots identifies as a transman or whether they're just not trying to be barbie.
Gender is stereotypes. And to the extent that all stereotypes are reflective of observed patterns, that holds. The problem is that sex is NOT a stereotype, it's an objective biological reality.
Black people liking rap music, fried chicken, watermelon, and sagging jeans is a stereotype. It's common enough among black people that it forms a pattern, but it's far from being causal, or even predictive. There are lots of black people that like classical music and steak and apples and suits; similarly there are plenty of white people who like rap and fried chicken and watermelon and sagging jeans. In that sense, "blackness" is analogous to gender.
On the other hand, black people have a higher melanin content in their skin, a predisposition toward extremely tight curls in their hair, and a materially higher prevalence of sickle cell disease and susceptibility to diabetes. Those are all objective biological realities, tied to genetic traits associated with descent from african ancestors.
It might be reasonable in some contexts to say that a white person "identifies as black" because they like rap and fried chicken and watermelon and sagging jeans. But it would be entirely unreasonable to say that such a person's psychological identity therefore makes them black in reality, and therefore entitles them to private clubs and scholarships that are exclusively intended for biologically black recipients.
I think we can ALL understand that distinction between an affinity with a stereotype and actual biological reality when it comes to race. Sex, being actually binary, is far less wishy-washy than race is... but somehow we keep ending up with people demanding that an affinity with sex-based stereotypes should override the reality of sex and allow males to access female single-sex spaces and sports.