Kind of repeating what has already been said, but perhaps in a slightly different way: It might help to see "sex" as the strictly physical/biological/objective side of the equation, and "gender" as the conscious/subjective side. (Objective as in a
physical object to be observed, and subjective as in a
subject that experiences and observes.)
Sex is what parts you have, and all the other physical characteristics unique to one sex or the other. For example, genitals, grip strength, facial hair, overall build, brain differences, hormones, etc. Gender (to borrow a phrase from Tom Nagel) is
what it feels like to be a man or woman. That brings in elements of behavior, roles, and consciousness/subjectivity.
Many specific male and female roles and behaviors are culturally conditioned, but on a more general level, the roles seem rooted in biology. Similar to language, the unique sound combinations of which are conditioned by history and culture, but which are rooted in meanings shared by all humans (for the most part - some words don't translate, but overall translation is possible).
For example, biological differences seem to influence many behavioral gender role differences, from temperament to mating strategies, among many others. And the Swedish experiments Peterson talks about come into play here too.
That applies to 99%+ of the population, with some variations. For example, not ALL women are more agreeable than men, but statistically, they're more likely to be. There are always outliers on any individual trait. Also, not all women marry men at or above their socioeconomic "level", but the vast majority do.
In other words, the vast majority of men (straight and gay) share an overall experience of "what it's like" to have male biology - same for women and female biology - even accounting for individual variations on any individual trait that collectively makes up the identity of male or female.
A tiny percentage, for some reason, despite having male parts, do not "feel like" a man. They feel "what it's like" to be a woman, like they're trapped in the wrong type of body. Interestingly, many children who feel like this grow out of it.
Some food for thought: a paper by Jim Tucker, reincarnation researcher, on gender identity disorder and reincarnation cases: https://med.virginia.edu/perceptual-studies/wp-content/uploads/sites/267/2015/11/REI29.pdf
... Stevenson (1987) has reported that the children he has studied who
claim to remember past lives as members of the opposite sex frequently
show traits that are characteristic of the claimed former sex. In one report,
he noted 121 cases of children claiming to remember lives as
members of the opposite sex, but he did not say how many showed
cross-gender behavior (Stevenson, 1986). In another, out of 34 children
that he had studied who survived past infancy and were said to have had
previous lives as members of the opposite sex, 21 (62%) of them
showed in childhood marked behavior appropriate for the opposite sex
(Stevenson, 1997). These cases have generally involved children in cultures
with a belief in reincarnation, and it seems extremely unlikely that
so many of them could develop this rare disorder independent of the
past life beliefs. Stevenson also says, however, that he has followed a
number of these cases as they have gotten older, and the majority,
though certainly not all, eventually stop cross-dressing and accept their
anatomical sex. Therefore, these cases may represent a variant of GID
that is ultimately less pervasive than in other cases.
Note that in all these cases, the issue is still binary: two sexes, two "what it feels like"s. No "third gender".
And it's not socially constructed. It is socially mediated, socially influenced. But society just colors the picture, it doesn't draw the lines in which the colors are placed. (Read the paper above for more interesting anecdotes about just how hard it is to "socially construct" a gender identity contrary to the one determined by sex.)
If sexual orientation is a result of imprinting onto biology, perhaps there is a similar imprinting that goes on with "gender", but if so, it seems even stronger and more effective than sexual orientation. Which makes me wonder if reincarnation trauma might be a good explanation for cases of persistent gender dysphoria.
And as for people who "identify" as a non-gender, or a hippopotamus, or whatever, I think that those examples are probably a combination of pretty severe psychopathology and just affectation, which I'd argue is a much better candidate for being a social construct.
) and what I observe is a profound hatered for life, family, values, and for humans etc. Often pop up something about mass sterilization, that children should be removed from their parents to be trained under state authority, and such things all while virtue signaling about feminism (the pathological kind, not the normal one that we all share) and talking about liberty, science (scientism in fact) and rights. Just an example: despite the horrible life women in places like Saudi Arabia have, all they're concerned about these days is that some people are complaining about some people complaining that some new movie being screened only to women (what? they assumed they gender?) and that it's wrong to say that it's sexist or whatnot. Who cares? People are suffering, people are dying, and they care about some stupid movie for preteens? The problem is that we can see these individuals as harmless but they are not. Underdeveloped individuals, when possessed by a destructive ideology can become violent when the opportunity is laid out, because there is no inhibition or self-restraint, those circuits have not been developed. 
