I guess I don't get non-binary gender?
This seems to be your core issue. It probably would have helped if you had made this clear initially. Probably, you're not the only one in this thread who is confused. Rather than assuming you have all the answers, or that the answers aren't important, maybe try asking some questions with an open mind. What is it about this particular topic that makes that so hard for you? Why do bring so much baggage to it? (Note: it's not just you. This attitude you have is very common, with this topic.)
Many people who use "they" as a personal pronoun don't feel that they are (for example) a woman in a man's body. Is it fair to say that they just aren't comfortable with who they are (regarding non-binary gender) or is it that they may be comfortable but just don't see themselves as a man or a woman?
The ones who are uncomfortable with who they are are generally not the ones who will ask to be addressed as "they". Think about it.
I though gender was mainly a cultural thing.
Gender and biological sex are interrelated, for obvious reasons. But yes, gender is largely in the mind, whereas sex is in the body.
If you feel that you are a woman in a man's body, that deserves a medical distinction.
You see it as a medical condition - perhaps one that can be or needs to be "fixed"? Why?
I guess I don't get non-binary gender?
Why do you insist that a person must identify as either male or female? If you can answer that, maybe you'll start to "get" what non-binary means, and possibly start viewing it in a different light.
If gender is cultural, why label it?
Is this so hard? Gender is usually linked with biological sex. There are obvious biological differences between biological males and biological females. Language has evolved to habitually mark those biological distinctions. The problem is that language has largely ended up pigeon-holing every person into one of two boxes labelled "male" and "female". While this works just fine for the majority of people, it ignores the feelings and identity of a significant minority of the population. It is an over-simplification of a more complex reality. It's a problem because its effect is often, in the end, negatively discriminatory against the minority.
Many people may get married and have a "traditional" wedding. Others may just go to the justice of the peace as a legal requirement and not really care about the ceremony. Others may just live together and not get married. Those people just aren't as "traditional" in that culture.
We don't try to have a label for them as a person.
We do! We especially have labels for women in the various states of marriage, because marriage has always been, at its root, a method to control women. Thus we label a married woman "Mrs" rather than "Miss" (and note the very recent addition of "Ms"; think about that). Women who "just live together and not get married" have traditionally been given rather more derogatory labels; I'm sure you can recall some of them. "Non-traditional" women (when it comes to marriage) have traditionally been looked down on and discriminated against.
If someone is born in a male body, is attracted to females, doesn't discuss sports frequently, wears khaki pants and cloth jackets instead of jeans and a leather jacket. Is that not a male gender. Is it non-binary? No, of course not.
Why "of course not"? How can you possibly judge, without asking the person how they identify in terms of gender? You're just making an assumption that you can put them into your "male" pigeon-hole. Why do you need to do that? What if, as well as being attracted to women, the person in the male body is also attracted to males? What if the person in the male body prefers to refer to herself as a woman? Why is this a problem for you?
What exactly is a non-binary gender and why do we even need a label for that? I have googled this and most articles spend all of the effort describing what to call them and not why we should call them something different or what exactly is different.
A non-binary person does not identify as either "male" or "female". They do not consider themselves to fit neatly into one of the two boxes you will allow people to fit into. They reject your system of classification.
Tell me why your system is the one we should all adopt, rather than respecting the wishes of non-binary people. What makes you King of World? What gives you the right to tell other people they must choose male or female, or (worse) that you will decide
for them what box they will be in?