The problem with any assessment of behaviour is that it’s impossible to say whether differences are innate or a result of socialisation. There are certainly feminists who have stated, here and on Twitter, that all behavioral differences between men and women are caused by the latter.
The reality is that we don’t know, as socialisation begins the moment we are born, so there is never a “blank state” situation and there is no ethically appropriate way to check. If however, we extrapolate from other mammalian species, it’s apparent that there can be quite significant behavioral differences between males and females, which are probably largely down to hormonal differences and which become more obvious after puberty.
The most obvious behavioural difference in humans, mainly because it’s the only one we tend to measure, which as far as I know has some consistency in various cultures around the world, is in violence and criminal offending. Some of that will be cultural, of course, but I doubt you will find anywhere in the world, where the women are more violent and commit more criminal offences than men, which suggests to me there is some innate difference, even if that is caused or related to physical differences in size and strength.
Regarding the nationality analogy, it works up to a point. I’ve often thought getting a GRA was a bit like me moving to a different country and working through the process to obtain a passport and citizenship. I’m then technically that nationality, but as I wasn’t born and raised there, I’m not really of that country in the same way those born there were. So legally I would be Spanish, for example, and the government would treat me as if I was Spanish. I might even pass as being Spanish, but I would know I wasn’t really, if I was any kind of rational person.
However, the analogy only works up to a point. Unlike sex, there are no consistent physical markers that would mark me out as British rather than Spanish. You might be able to instigate some kind of DNA testing, but even if you found I was ethnically Spanish, descended from families that had lived in Spain for millennia, that wouldn’t make me Spanish necessarily because nationality is effectively only a legal label, defined by where you were born or have lived or where your ancestors were born or lived. Being male or female is physically dictated by genetics and generally very easy to identify. It’s not simply a matter of legal labelling, although that is what gender ideologists are trying to turn it into.
As to what gender or gender identity is, gender ideologists seem to be pushing for some definition of it being “an innate understanding of what sex you are”, but I would argue that “having an innate understanding of what sex you are” ought to be closely linked to your actual sex. If you believe yourself to be the opposite sex to reality, then you probably need psychiatric help because something has gone wrong. Which is why the concept of “gender identity” needs an entire new, invented language to back it up. Without that language and the faith to believe something unproven and unprovable, the entire concept falls apart.