I think "gender identity" doesn't need to be a thing. To me, gender is just a classification of what bits n' pieces you got. Whether you feel more girly or manly is just based on the individual, so why try to conform to a title?
Giant wall of text, there is a tl;dr at the end though.
It's actually been shown that people who do not identify as the gender they are assigned feel real psychological stress from being called the incorrect gender. I mean, I suppose it's hard for me to understand, because I have not had questions about my gender, but imagine it as if you identified as a female. Now gender is a fluid thing, but some people do exclusively identify as a female, not a male or not a bigender, but specifically a female, and in that case they still can't do a lot of things a normal female can do. One example that is often cited is that a transgender female (that has the sex of a male) has to use a male restroom, it seems kind of a trivial aspect, but there's a lot of psychology that goes into it. You feel alienated from your physical gender and at the same time not accepted by your mental one. Some people truly don't feel just "girly" or "manly" but they truly identify as a girl, whether it be the spitting image of the social constructs of the girl, or their brain is actually wired differently. Regardless, they feel that way [to such a degree] that some are willing to change their entire bodies in order to have their physical and mental gender match. The main thing is that it's not just feeling manly or girly, but it can be such an important aspect for people (and it's not very possible to make it feel not important) that these labels can become more important. Gender is pretty much the same as sexuality, in terms of this regard, and while some straight people can call a guy on TV handsome, there's still a fundamental difference between that sort of non-sexual idealistic attraction, and the sexual attraction that a gay person can feel.
People like labels. People like to be called a nerd, a jock, a cool kid, a tall kid, an American, a gamer, a furry, a brony, people love labels. The reason people love labels is relatively simple, it makes people feel like they can belong. If a label exists for you it means other people match that label, which means you're not alone. There is a contrast to this, where some people try to exclusively fit OUTSIDE of a label, but usually these people STILL fit into labels (notice anarchists tend to stick together) but in general most people like to fit into labels. People like to feel accepted and normal, and if there's no label to identify yourself with, you can feel alone, separate from everyone. If you're a transgender you aren't a man and you aren't a women, because it depends how you define male and female, so without the label of transgender, you feel alone. I'm an advocate of simple labels though, I think it gets a bit too far when you exclusively try to call yourself one thing that is nearly the same as the other, but in the case of transgender there isn't really a label for that.
So basically, tl;dr. Having a differing gender to your sex is not just believing you are manly or girly, it is instead something much deeper, it is comparable to the difference of a straight person admiring a male body and a gay person doing so. Labels are important because they allow people to feel they belong in a community, almost everything is in a label, such as being a gamer, if you do not fit into a label you may feel alone and so people like being able to fit into labels.
Do I agree with the classification of labels? No, ideally people wouldn't worry about that. But I can definitely understand why people do.