Yes, gender identity is really important socially. We are constantly being "gendered" right from infancy, and sometimes even while in the womb. But gendering is a social process, which means that it is learned behaviour that responds to systems of thought and cultural practices all around us.
Biologically speaking, sex is a huge spectrum. Most people who study it medically would tell us that there are a range of physiological as well as biological characteristics that are in between what is technically classified as female and what is technically classified as male.
Some feminist theorists (e.g. Judith Butler
) have argued that gender behaviour is so important socially, and that we are so emotionally committed to the supposed binary nature of gender, that we have mapped learned gender characteristics (our gendered behaviour) onto biology. So her argument is that we map gender onto sex, just so that we can believe that sex is binary. It's an argument that has merit, especially because biological data does back that up, as I mentioned above. Most biological data that explains behaviour through purely genetic data is fairly flimsy and a lot of questions remain about why that kind of research is important.
(I won't comment here on the "gene industry" and what kind of research fund motivates that kind of research - there is an awesome book out there called Biology as Ideology by Richard Lewontin
which lays out a compelling argument, backed by empirical data, about how ideology and societal forces - such as capitalism - influences "cold, hard" scientific facts that we believe to be based on unbiased truths).
Then the question remains about how and why we stick our learned gendered traits. The answer to that, according to Butler, is that we perform
gender all the time in response to the world, kind of like drag to take an extreme example. It is part of our social existence. Some other feminist theorists have taken some issue with that, saying that Butler's arguments about performativity make it seem like gender is a choice, and it is a lot more complicated than that. Words like gendered
draw attention to the fact that these are social processes, and we don't have a choice, as an individual, to necessarily resist social processes that occur to us/through us.