Isn’t it obvious that the halakhic “woman” means a biological woman?
Back then there was complete identity between the two (gender woman and biological woman), or perhaps more accurately, there weren’t “two” categories at all—there was only biological sex. And the new sociological distinction that creates a category umbrella for people with traits usually identified with biological women is not relevant to the halakhic woman at all.

Is there any ontological connection between trans people and a biological woman?

To me that is really not clear at all. On the contrary, reason suggests that “woman” in Jewish law is determined by gender and not by sex, because why would halakhic matters depend on physiological structure? It is far more reasonable to tie them to psychological structure, which is more essential. Even with signs of maturity that are tied to physiology, it is pretty clear that those are only indicators of the essential signs (which are hard to examine and define with sharp criteria).
True, in the past the Sages never imagined such a distinction, and it was obvious to them that sex and gender went together. But in the last generation, the understanding has developed that this is not so. Sometimes there is a split between sex and gender (see column 504). So now the question arises: what was each law really dependent on in the past—sex or gender?
This is a halakhic change similar to Meiri’s remarks regarding gentiles. In the past, every gentile was presumed to be wicked and an idol worshiper. Over the generations, a distinction emerged between those parameters, and there were gentiles who were not idol worshipers and were not wicked. Now one must ask whether the halakhic attitude toward gentiles depended on their being gentiles or on their wickedness. One could have asked that even then, but as long as there was no split, the question did not arise.
Something like this is also true of the laws of custom. Originally, custom was determined by place. But in the ancient world, which was much less dynamic than ours, ancestral custom and local custom were the same thing. People usually lived where their ancestors had lived. In the new, more dynamic world, a split occurred and place is no longer so fixed. If so, ancestral custom no longer overlaps with local custom. Therefore a halakhic dynamic emerged in which local custom was replaced by ancestral custom. Rabbi Ovadia argued that in the Land of Israel everyone should follow the local custom (the place of Maran), but in practice that was not accepted. Everyone follows ancestral custom (Ashkenazim and Sephardim). And likewise with “do not form factions,” which contains a prohibition (plainly, on the Torah level) against maintaining two synagogues in one city, and nowadays no one even thinks of complying with that. The city as a location has become an irrelevant parameter. For the same reason I argued that one can join a prayer quorum on Zoom. Physical geography is no longer significant. The site (the virtual one) has replaced the place (the physical one).
Changing reality requires a different application of ancient laws. Exactly like the example of swimsuits.

Why is what you wrote about “a man shall not wear” not deriving from the reason for the verse? There is an explicit prohibition on a man wearing women’s clothing, and you are pouring into the word a content that did not exist then, based on your own reasoning that the Torah would not tie laws to physiological structure. But the reality is that it did. There could be many reasons for this: maybe dressing according to gender rather than sex is itself licentiousness. Maybe it does not recognize the possibility that there can be a difference—that is, a person’s internal feeling has no legal significance regarding how we relate to them. Maybe, as Maimonides says, the Torah knew that the commandment would not be “correct” for a negligible percentage of the population, but that is true of many commandments.

But even before that, there is an incorrect assumption here. The content of the words “man” and “woman” has not changed, except among a tiny fraction of the population. Most people may recognize the right of an adult to do as they wish, but they will generally still relate to a person according to their biological sex (not to their face, so as not to hurt them, but for example they would not enter into a romantic relationship with them). That is not the situation regarding gentiles, where reality really changed, and also not the issue of the site (where I disagree with your conclusion, but I understand the reasoning).

Not at all. The question is what kind of “woman” the prohibition is referring to, not what the reason for the prohibition is. The Torah indeed forbids a man to wear women’s clothing; the question is what the “man” is that the Torah is talking about. You assume that it tied this to physiological structure, and I do not know where you get that from when reason suggests the opposite. Notice that I am not changing anything in the commandment; I am only interpreting its meaning from the outset.
The content of the words you are talking about concerns terms—that is, words in language—whereas I am talking about concepts, meaning what the words describe. Unlike words, the meaning of concepts can be different even if most of the public is not aware of it and does not agree with it. If the truth is that a woman is a matter of gender and not sex, why should I care if most of the public does not think so? And if later most of the public does think so, then Jewish law would change? By the way, I am also not sure you are factually right. I haven’t checked. But as I said, that doesn’t matter.

Assuming you’re right, would it then come out as a stringency that it would be forbidden for him/her to wear men’s clothing because of “a woman shall not wear”?

Absolutely. But I wrote that I’m not definitive about this, and for me the situation remains uncertain. So there is room to be lenient or stringent either way.

Assuming gender is what determines it, why does the Talmud discuss the definition of a tumtum and an androgynos? We could simply check which gender they feel they belong to, and on that basis determine whether the person is a man or a woman.

Strange question. Before you even get to a tumtum and an androgynos, ask why we go by biological signs at all in order to determine whether someone is a man or a woman. But beyond that, I already wrote that there are laws for which sex is what determines things, not gender. And besides, it is obvious that the Sages did not see it this way. I am talking about changes in understanding, not claiming that this is how the Sages thought. Was it for nothing that I brought the examples above about changes in Jewish law?

If I understood correctly, you explained that the biological signs indicate psychological maturity and something essential, and that is very plausible (though even there one could ask what happens if someone has the signs but psychologically is still a minor?).
I’m asking about a tumtum since presumably they feel belonging to one of the genders (or perhaps they feel belonging to both genders together?), and if so, shouldn’t their status be determined by that belonging?
For example, regarding the fact that they cannot discharge others’ obligation because it is a case of doubt.
I understood that you do not mean to disagree with the Sages; you just said that in their time they did not recognize such a reality, but if you had asked them, they probably would have had something to say.

Indeed, but those are still only signs. My claim is that it may also be the case that the physiological signs of maleness or femaleness are only indicators of the real, essential thing.
I answered your question about a tumtum and an androgynos.
What I mean is to say what is true. Whether that disagrees with the Sages or not is not really interesting. When you are not aware of such a distinction, you have no position about it. The question of what they would have said had they known current information is not well-defined. If they had known current information, they would not have been the Sages but people of our generation. The question is what is correct for the present generation, not what Rav Ashi or Maimonides would say today.