gender is just a linguistic category. Humans also have a habit of categorizing things by sex because it's kind of an important part of our continued existence, so you ended up with "male", "female" and "other" groups in a lot of languages: he, she, it. Gender and sex are used interchangeably there: gender is just a reference to the sex.

The basic definition of sex is pretty simple and tied to reproduction; there are different kinds of people who have different roles to play in the act and it doesn't meaningfully refer to anything else. However, complex definitions are different; people have a lot of ideas about what it means to be a certain sex: how they dress, act, what they think, etc. While they may have some merit as general statements(studies show a majority of men do X, etc), as anything beyond that, they're simply incorrect. When taken as a definition of sex, they basically represent a superstitious understanding of what sex is. We have the basic understanding of sex, we have the scientific understanding of sexual development, and then we have a bunch of bullshit attitudes about sex people insist on layering on top of all of that.

Societal gender is the bullshit. It's what you get when someone sees all of the confused ideas about what a male or female is supposed to be and gives that priority over sex. So if someone doesn't fit their gender stereotype, you don't say "see, the stereotype is stupid", you say "they're actually a different gender." There's no real definition of gender below the mess; no clear definition of what the male/female/other genders are; no way to look at someone who says "I'm X gender" and show them how they're incorrect. Even "gender identity" is defined as "the gender someone identifies as"... it's not what they are, it's what they call themselves. The desire to be a certain gender takes precedence over it actually meaning anything, because a meaningful definition presents the possibility that someone's belief about themselves could be wrong.

There doesn't seem to be much good philosophy behind any of it, it's just a lot of bad philosophy people politely ignore. The bad philosophy is also how you end up with people identifying as different races or lawn chairs or whatever. It's stupid, but the arguments hold up... however, since people don't really care about the actual arguments, they just stop being polite for a moment to tell those people to fuck off.