Okay - warning, massively long post ahead (but I don't think it can be done in any shorter way). I'd like to thank argumentativefeminist for giving this a go, when most on her/his (I don't know how he/she identifies) side of the argument don't try, and I'm going to quote this and try to "translate" it for those who prefer their philosophy in "plain language" (many aeons ago I was an academic philosopher, albeit in the Anglo-American analytical tradition rather than the continental one).
Following Foucault, though he doesn't speak on gender specifically, and Butler, gender is a concept created by the institutions of society to fit it's needs. Hence "woman" emerged as a way of classifying people with vaginas. There's a couple of arguments here of why we shouldn't follow this classification. One, a surprising number of people are born intersex - How can we classify them into a gender category? Surely, we should really be allowing them to choose what feels most natural to them. So why cant other people do that? Secondly, and more abstractly, the relationship between signifier ("woman"/"tree") and signified (👩/🌳) is an arbitrary one. Plenty of signifier/signified relationships change over time and many feminist linguists (e.g. Cixous) advocate for challenging these relationships where it seems that they benefit or uphold patriarchal institutions. I would argue that the gender binary as it was originally conceived benefits patriarchal institutions, so I advocate for altering the relationship between "woman" and 👩 to include people who don't fit the traditional physical characteristics for "woman". At the end of the day, I can't see that this in itself causes any harm, although I understand that some people have concerns about certain legislations etc., but I dont think thats ultimately what's being debated here.
The way I used to gloss Foucault for first year undergrads was by way of a quick and dirty analogy: Both Bacon (16th/17th C) and Foucault (20th C) were fascinated by (among other things) the connection between scientific knowledge and political power, but there's a sense in which they looked down the telescope from opposite ends. For Bacon, possessing knowledge gave/reinforced political power (make better gunpowder and understand the mathematics of ballistics and you have better canons for your army). For Foucault, having political power enabled you to define what counted as scientific knowledge (e.g. define mental illness in such a way that you could incarcerate your political opponents in mental asylums).
So that's one part of the story - for Foucault and writers who follow/borrow from him, knowledge isn't absolute, it arises in a social context, serves political ends, and can be changed according to political needs.
The second strand to the story (I think) is that language can be seen in a multitude of ways. It clearly partly helps us to describe and talk about the external world - I want to tell you there's a wolf outside the cave, so you'd best not go out. But it also fills a social and political function - it is shaped by, and shapes, how we can talk about political alliances, power imbalances (and indeed ways in which we can't talk about them - if you don't have the words, you can't say the unsayable - something that arises time and time again about whether oppressed groups have the necessary language to name and talk about their oppression, or whether that language has been taken away from them by the ruling class).
So I think what argumentative is trying to say is that "woman" isn't just a word labelling an uncontentious biological category (large-immobile-gamete-producing member of the human race with the ability to gestate and lactate offspring), it's a political category. Politics/social power imbalances told us why we needed the word in the first place (why not just "human"?) and attach a whole load of other baggage to the word (gendered expectations about behaviour, dress, place in society) - and in fact (following Foucault) there's no independent "scientific" real-world meaning which exists in some sort of pure form, stripped of all the extra political baggage.
Therefore (and here's the real sleight of hand in my books) the word is up for grabs to be redefined in what I personally see as a political power grab - people with power (middle class white males with a university education which gives them access to po-mo word salad) are now free to cut the word completely adrift from its reference to "pertaining to the class of humans capable of gestating and lactating" and leave only the extraneous political baggage of performance of femininity, submission, subordination.
I'm not sure argumentative is entirely consistent in her/his approach, however, as this sort of Foucauldian analysis relies on understanding language as a social phenomenon (correctly so, though I think that questions of what in the real world language refers to are more important than some Foucauldians believe). However, in later posts argumentative seems to go with the idea that this all works at at the level of individuals, not society as a whole, which is wrong both as an understanding of how language works, and in terms of understanding the practical consequences.
For instance, argumentative holds that it really is possible for someone born biologically female to completely identify out of womanhood. But this doesn't work at an individual level - see for example the tragic and awful case of the poor transman who was raped by a taxi driver, and in the subsequent police interview said "but I kept telling him over and over again that I was a man." Nor does it work at a political level - see the thread on "why can't transmen inherit a peerage?" Those with political power make damn sure they keep it. So, self ID fine for anyone identifying "down" the power ladder into womanhood, not so acceptable for people trying to identify "up" the power ladder into manhood. And what we see very clearly here is that the social consensus view of meanings ultimately trumps the individual assertion view of meanings. Very much at the level of "war is diplomacy by other means."
So who benefits from allowing people to pretend that this is happening at the level of individual assertions about one's own identity? The answer is pretty clearly men. And who loses out by having the language needed to describe their oppression stripped away? Again, pretty clearly women.
The Foucauldian analysis, it seems to me, is right, but not in the way argumentative thinks. It's right because it articulates the way language gets co-opted into power struggles on a society-wide level. Where argumentative goes wrong is to think this gives a licence to apply this at the level of individuals making assertions about their own personal identities which then are completely anchorless - not rooted in the external world, not rooted in social consensus about how we should categorise that external world.
And of course that's what the mantra of TWAW is supposed to achieve - it's a blunt instrument to artificially construct a social consensus. Equally that's what makes it so politically important for women - old-fashioned cunty women - to resist the artificial construction of this social consensus.
Top marks for staying power if you made it this far, and apologies for the wall of text, but I couldn't think of any way of condensing this down!