This post is primarily directed towards trans people who find the idea of socially-constructed gender invalidating. The reasoning typically goes sort of like this:
If gender were a social construct, then that means that I'm only the gender I am because of the society I live in. But I don't feel like that's true. I feel like I was born wanting to be my gender deep down, and even if I lived in a society without gender roles or stereotypes, I would still want to transition to be my gender. There's even some scientific evidence that trans people's brains have certain similarities to those of cis people of the gender they identify as. Therefore, while gender roles and stereotypes are socially constructed, gender itself is not: it's innate.
It's probably true that people have inborn feelings that cause them to identify with a certain gender. However, this is entirely consistent with gender being a social construct. To see how consider the following:
Growing up, Fern always felt detached from the gender they were assigned at birth. Sometimes they thought about being the opposite gender, but really, they felt uncomfortable with the whole idea of being a boy or a girl. When they were a teenager, they learned that some people are non-binary and quickly determined that their feelings lined up with them being non-binary too. They started going by they/them pronouns and dressing in a way that kept strangers guessing about their gender. Eventually they even went on a low dose of HRT and got an X on their drivers license.
Now, let's rewind time to when Fern was a baby. Let's assume that there is something about baby Fern's brain that made it so they would grow up to have the feelings they did and end up identifying as non-binary. Then, let's take baby Fern's brain, rewind time again, and put the brain in the body of a baby born in the year 1900. We'll call this baby Fern-100.
Growing up, Fern-100 always felt detached from the gender they were assigned at birth. Sometimes they thought about being the opposite gender, or even some sort of sexless being, like an angel, but they knew both of these were impossible. Throughout their life, they dressed in a somewhat more androgynous style than most people and were often suspected of being a homosexual. In their 50s, they read a newspaper article about a man in Europe getting surgery to become a woman. They found themselves strangely fixated on this story, but knew that the operation wasn't for them and that they would spend the rest of their life as what they born as.
Fern and Fern-100 had the same innate detachment from their birth gender, but only Fern transitioned or identified themself as non-binary. The difference was the society they lived in. The concept of being non-binary was simply not available to Fern-100 because of the way their society constructed gender: not gender stereotypes, not gender roles, but the basic categories of gender itself. That's what (sophisticated) people are talking about when they say that gender is socially constructed. They're not saying that people don't have innate tendencies related to gender; just that the results of those tendencies depend on the society they live in.
You might wish to say that gender really just refers to the innate stuff, and that socially constructed gender categories are a different thing. But I think that's problematic for a few reasons:
Historically, the distinction between gender and sex was made specifically so there would be a label for the socially constructed part, and that label was gender. It's confusing to use a definition of gender that comes pretty close to the opposite of the one most sociologists use. It would be much clearer to use a separate term to refer to the innate feelings that cause people to identify with a particular gender. Julia Serano calls it "subconscious sex," which I think is much clearer.
Regardless of whether they're trained in sociological theory, people usually refer to socially constructed categories like "demiboy," etc. as "genders."
There are a lot of societies out there that have different genders than the ones in modern Western culture. For instance, most Polynesian cultures have a third gender for feminine AMAB people. My understanding is that some of them consider themselves to be transgender in the Western sense, but many do not. If we say that gender is innate and has nothing to do with culture, it comes pretty close to saying that māhū, fa'afafines, etc. are actually just the same thing as trans women, regardless of what they think themselves, which is pretty colonialist.