Can you elaborate on what makes that statement so ridiculous on it's face?
What does it mean for these categories to be social constructs? What criteria makes one a man or a woman?
That's up to the society, hence their definitions as social constructs. For example, when you shop for dresses, do mostly men or women come up? Articles of clothing by themselves are not tied to the biology of a person's sex... meaning a vagina/penis is not required to wear a dress.
But most of society (as it is now) has deemed that woman are associated with dresses, while men are associated with suits. That image is now changing, although slowly, with trans and other non-binary genders.
An easy way to see this is to ask yourself if you think it's acceptable for men to wear dresses? If so, ask why we don't see more of that in the workplace.
So, if someone identifies as as man, but society disagrees, is that person a man? What reasons would a society have to change their criteria to include this person in the category of "man"?
Marriage used to be strictly between a man and a woman. Now society says it can include homosexuals. Women used to stay at home to take care of the house, while men worked at jobs to provide for the entire family. Now society says both roles can be taken up by both men and women. Black people used to sit at the back of the bus, and now anyone can sit anywhere.
I mean, society constantly changes. We see it in history, we even see it today within our own lives. Places like Saudi Arabia are currently having their own version of woman's suffrage even as we speak.
So given this, in your example person, I would wager that society would deem that person not a man, since the hypothetical society has already decided it that way. But that's not to say that person will give up on not being recognized as a man. Our human history has shown us that we don't just stop at an idea, some of us go all the way to pave new rights for entire future generations to come.
Whether or not society adopts the new definitions, well, that's up to the people living in it.
The Wikipedia article on Sex Differences in Humans uses the terms man/woman and male/female interchangeably[1]. What little I've seen of the scientific literature follows this convention as well.
It's my experience that, across a broad swath of American society, that many people follow the old convention as well.
Yes, in most contexts in present western society, male/female and man/woman are interchangeable (and certain groups have a vested interest in maintaining this state). There are however contexts in which they are not. Trans people (and allies) need to be cognizant of these things to be able to discuss the differences.
If your argument is that to an observer, "trans women are women" could be interpreted as "trans women are biologically female", then sure one could interpret it that way. But in the context of discussions about trans people, women and female mean different things, which is why "trans women are women" is the phrase, and not "trans women are biologically female".
That dictionaries haven't caught up is kind of disappointing, but if you look at the wiki page, it mentions trans women at the end, which seems kind of strange to do in the context of biological female-ness.
I'll also note that because it's so common to conflate the two, that I try to explicitly add the "biologically" modifier when discussing bio sex. in contexts not talking about trans groups, I'm sure I've conflated the terms without noticing.
In terms of the science of sex differences, my understanding is that they (at least sometimes) don't make the distinction between sex and gender because many traits and social behaviors that we might assume to be part of "gender" end up having a underlying biological component as well (that isn't easily separable).
In terms of the general populace, I would say that for many people the realities of their biological sex (including their reproductive capacity) have a great deal of impact on their gendered experience.
This is not to say that the experience of trans people should be ignored. But this may help account for some of the surprise people express at statements like "trans women are women" or "women is not a sex".