Why do cis people need the concept of "biological sex" so much? Why do they have such a strong need to put trans people in their place by saying, "Sure, you identify as a woman. But your biological sex is male"?
At the root of cisnormativity, like all other harmful normativities, is a desire to control. To exercise power over somebody else. And telling somebody, "You aren't really who you say you are -- I can invoke some greater authority that says you're lying about who you are" is a way of controlling somebody else. It frames that person as an unreliable narrator of their own experience, and reinforces the cis person's greater power to name, to identify, to categorize.
It doesn't help that the watered-down liberal version of trans education that has been promoted for a long time emphasizes the difference between "sex" and "gender," making cis people feel like they can evade criticism as long as they memorize that talking point. It also doesn't help that anyone who challenges the simplistic sex/gender binary gets accused of wanting to alienate allies or wanting to make it harder for them to understand us.
That didn't cause the problem, though.
"Sure, you identify as a man, but you'll always be truly biologically female" ultimately means, "What you identify as doesn't matter. It's not real; it's all in your head. My objective observation of your body is that you are female, and that's scientific."
There is no rule of science that says we must use terms for other people that they wouldn't use for themselves. That's social and political.
So the attachment to "biological sex" is really about saying this: "There is something other than your own self-description that I can use to classify and categorize you without your consent. I can categorize and label you based on externally observing your body, without asking you what categories you belong in." The power to name is the power to control. And cis people react badly when we try to take this power from them by saying that "sex" is just another name for gender.
It's easy to see what purpose "biological sex" serves structurally: gender-based oppression would no longer be possible if gender categories were entered into consensually. To oppress somebody, you need the ability to place them in an oppressed class in a way that others will generally recognize as valid.
But on an individual, psychological level, I wonder what purpose it serves. Why is there such a strong need in so many cis people to tell somebody else they're wrong about their own sex?
One answer is that cis people don't like admitting mistakes, and that most cis people learned as children that boys have penises and girls have vaginas. When faced with a choice between recognizing trans people as fully human, or maintaining their own omniscience, they go for the narcissistic choice of refusing to admit that what they learned early on was incorrect.
But I don't think that's the whole story. People make all kinds of mistakes, but admitting that they were taught something incorrect about sex categories seems uniquely difficult.
So I'm leaving it here as a question. Why does any individual cis person feel such a strong need to tell a trans person, "You are truly biologically male," or, "You are truly biologically female" when that isn't how the trans person would describe themself? The answer isn't "science", since science doesn't require anybody to place others in particular political categories; as well, very few cis people saying this have any understanding of science. I don't know the answer to this question, but I think the only way to begin finding it is to reject the pseudo-scientific notion of biological sex as objective truth rather than socially and politically motivated narrative. We have to stop asking what biological sex is, and start asking what work the concept of "biological sex" does and what needs it satisfies.