I think that it makes sense for there to be a biological model which divides humans (and some other things) into two reproductive groups, because reproduction essentially works by having two types of people which interact with each other. In practice, there are still going to be people which don't fit squarely into the two idealized classes though, such as intersexual people, homosexuals, asexuals, and etc. In this sense, I don't think its unreasonable to say that there are two sexes (male and female), but that some individuals fail to fall into either category (arguably because of a mutation, but the reason doesn't matter here).
Gender is also a very cultural thing. Animals possess some instinctual understanding of their own sex, I think, but the idea of sex is probably exclusively a human construct. A wild monkey might have thoughts or feelings resembling "i have x body part", or they might be able to look at a certain monkey and think "i want to do x with that monkey", and the sum of those thoughts/feeling might constitute most of that monkey's understanding of its sex. (although those simple thoughts/feelings could result in some relatively sophisticated emergent behaviours)
Humans have two poles when it comes to sexual specialization (male/female) but in practice sometimes regard sex on a social level as being more than two-pronged. Eunuchs are probably the most prevalent example, where in large parts of asia and europe historically if a child asked their parents "is that a man or a woman" they might get the answer "that's a eunuch" and if a child drew a picture and their parents asked if it was a man or a woman, the child might answer with "neither, its a eunuch." Biologists (or the equivalent) at the time might not even consider eunuchs biologically male, since they lack what they consider a mandatory part of being considered male, in the same sense that they wouldn't consider a baby who was born without certain parts as being male. There are some other historical and obviously modern instances where it is ingrained even among young children that there are people who are neither male nor female.
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Now what happens when this boy gets a little older and learns about biology / sex? If someone tells him "men have penises and women don't", he would tell them "but I'm a woman and I have a penis". Since he knows about gender before sex, if the two contradict, he would assume his sex is wrong rather than his gender being wrong. But his sex can't be wrong, because that's the body he was born with. A sex change is not a nice procedure to have to go through; it's better to treat this sort of condition with counselling to help him accept that he is a boy.
This is something that I feel i should specifically respond to. Counselling is not necessarily a nice thing to go through either, especially for a child, and there are a lot more options than just counselling or a sex change. I think the average parent would (and probably has in the past) just ignore it and wait for them to grow out of it. The reason we're starting to respond to it more now is because we're beginning to identify transgendered people as a real class of people and we've discovered that they have an extraordinarily high suicide rate, along with a host of other problems.
Because this problem is a real and very severe issue, I would strongly advocate that academic research be done on the best way to deal with the issue, and that anyone facing the issue of how to deal with a child showing these kinds of warning signs defer to the results of such academic research rather than making a decision on their own.
And as a closing note, I think that calling others how they like to be called, whether its a matter of name/gender/species/whatever, is an easy low-effort way to make someone happy and show that you care about them on a basic level.