Are you saying that identity is entirely free choice? Don't we portray ourselves to others in very specific ways precisely because there are standards for different kinds of identities? I can theoretically decide that my identity is all kinds of things but if it doesn't correlate with the expectations of others then I'm just going to be the only person who thinks that way. If I identify as incredibly attractive but I'm not, or as a particular ethnicity that I'm not, or I view myself as a badass but I don't train or exercise whatsoever, isn't that just me being delusional? If identity was simply whatever people thought of themselves, then there shouldn't be any problem, as nobody can force someone to stop describing themselves in a certain way.
The issue is whether others accept the identity you choose, and the question here is the legitimacy of a choice to determine one's own gender or what the prerequisites for being able to make that choice are. We definitely have at least some, gender isn't meaningless because it's recognizable, otherwise nobody would care about being a particular gender in the first place.
As you surely know, one of the problems with the "non-binary" idea is the role of gender in English. It's not just pronouns, many words have male/female variations that we commonly use. To speak everyday English requires us to identify people as male/female, and use the appropriate pronouns and language. Often, people who are non-binary don't want to be either gender, they want to be referred to using gender-neutral terms and pronouns, which requires careful word selection that is fairly unnatural to people, considering it is very rare to interact with someone who considers themselves non-binary.
Besides that, many concepts in society are gendered, such as how areas, activities or services are designated for the exclusive use of a singular gender. Cultural and social expectations are different based on gender, as are many different types of ideas. The vast majority of people do fit into one of the two gender categories, which partly explains why so many things are organised by gender. I'm not saying it has to be that way, just that it is.
Our language and organisation could be less gendered than it is, but so long as it is so gendered, it is impossible to argue that gender is "meaningless". If one is indifferent to their gender classification or agrees with it, then it's not too much of an issue. The difficulty is for non-binary people who resent being categorised by gender, preferring gender-neutral language and gender-neutral treatment. Who resent gender-specific norms or cultural or social expectations, and so on, because now this entire system has become problematic.
I agree that gender characteristics normally ascribed to one gender can be exhibited by someone of the opposite gender and that this shouldn't be a problem. However, it is wrong to say that there is no correlation between one's gender and one's sex. We know that there are many psychological and behavioural differences between the sexes and this has been proven in many different fields. Most people are comfortable with their gender being defined by their sex. Most people take no issue with their gender correlating with their sex precisely because of this correlation, it is a minority who don't. This difference is not just socially manufactured, it is the naturally occurring difference between men and women. I think it is wrong to try to force people who don't fit into these categories perfectly to do so, and that people should be allowed to express themselves without feeling hindered by gender norms. That's what we've been moving towards, and I think it's great.
Gender is a significant part of our language, culture and the way in which we organise our society. It's part of how we understand people and part of how we think. There is a small percentage of people who are non-binary, who did not easily fit into either category as most do. Based on all these things, leaving aside the issue of what should be or could be, and despite being imprecise, gender is definitely not meaningless.