If gender is socially constructed (and that is indeed how the term is defined today), then it's exactly as binary as we want it to be. We can change it, and indeed we do, all the time.
There are three genders in the German language for example, so there is nothing about the word "gender" which dictates only two possibilities. We don't need to remain trapped with a set of two, and there's plenty of evidence that slavishly echoing this narrow range causes all kinds of problems.
We could start by accepting the scientific fact that there are at least a dozen different biological sexes, so that we're not overly hung up on 'reflecting the two natural sexes in culture' or some other essentialist nonsense. (Intersex births are approximately as common as red hair, about 1 in 100). If sex is not necessarily a matter of only two possibilities, why on earth are people so anxious to propagate that fallacy into culture? So anxious to insist that people who (for whatever reason) fall outside this two-way split are abominations, or just need to "get over it". Have you seen the suicide statistics for transgender and intersex folks? We can do better.
Besides, gender norms are already fluid. My kid tells me that purple is a girls' colour and yellow is a boys' colour. It was certainly the other way round when I was a nipper. Astonishing as it may seem, it was pink dresses for boys right up until the 1920s. Most of the first programmers were women too. More hours are spent playing computer games by women than by men (true since at least 2003). A lot of assumptions about gender turn out to be either unreliable, or quite false on closer analysis. This research reflects that quite well, despite the best efforts of some commentards here to see it as some kind of feminist/SJW conspiracy.
FWIW I identify as 'non-binary', despite some discomfort about the nomenclature*. No I wont discuss my genitals, but I am happy to talk shop about software development any time.
*I dislike the way the term "binary" is used in the context of gender discussions, because, as any programmer knows, faced with variety limitations, you just add more bits. You can have all the richness you need if you go beyond a one bit system. "If a system is to be stable, the number of states of its control mechanism must be greater than or equal to the number of states in the system being controlled" - as W.R. Ashby so beautifully put it.