I think the best way to illustrate it is by thinking of the example of a trans person who ‘passes’. I’ll take the example of a trans man, and think about how they are a man in the social sense, as I know some people dispute that there are any trans women who really pass, whereas I think it would be difficult to claim there are no trans men who pass.
So this trans man is regarded as a man by all those he interacts with - they instinctively refer to him with male pronouns, and consciously or unconsciously hold beliefs and make assumptions about him as a man (which will be highly dependent on their own experiences and biases) - they might assume he is interested in sports, not interested in babies, finds it hard to express emotion, that he’s assertive or potentially aggressive. That he’s more likely to be a lorry driver or doctor than a ballet dancer or nanny. That he’s more likely to be attracted to women than men. Of course, it may be that these assumptions are all incorrect.
On the side of the trans man, he may or may not feel comfortable with all of those assumptions about him. But it feels ‘right’ when he is referred to using male pronouns - it feels like he is being seen as he really is (at least regarding his gender), whereas as female pronouns would feel ‘jarring’. Knowing that other people perceive him to be a man similarly feels right.
The fact that it is possible for this person to pass through the world of social interaction indistinguishable from any other man (even though it is clear on a close anatomical inspection, or through medical tests, that he is a trans man and biologically different from men who are not trans), suggests to me that there is a real social category of ‘man’.
Of course, there is then a separate question about who falls into the social categories of man or women (and whether some people don’t fall within either), and who gets to decide who is in which category. Not everyone who wants to fit with a category will be perceived by others as falling within that category. There is also the question of what significance is attached to those social categories - should it determine what people can do, what services they can access or what spaces they can enter? Or should it just dictate what pronouns are used to refer to them?