Asking someone "are you a man or a woman?" is as illogical and leading as asking "what is your favorite flavor of ice cream, chocolate or vanilla? Circle one." When you ask a question and specify only two possible answers, almost no one* (statistically speaking) will choose an answer not given. But this is how people ask the question about gender, if they ask it at all.
Also, if you ask people "what is your favorite flavor of ice cream? Check one: chocolate, vanilla, strawberry, praline, coffee, blank," you will definitely get more answers than just "chocolate" or "vanilla" but people will still mostly choose from the options given, even if that list doesn't make sense to them.
By framing the question in a way that makes being specific more work, you increase the barrier to being specific. Also, social desirability implies that anything on the list is more desirable than anything not important enough to be on the list.
So instead of making their own answer, many people will choose the one that is closest. For example, people whose favorite is rocky road may choose chocolate. Or maybe their favorite is a very unusual flavor that most people are unfamiliar with, so they choose the one that is closest while still being familiar to others. For example, my favorite ice cream of all time was Sheer Bliss pomegranate dark chocolate chip ice cream, but it is no longer in production and I have hated every other pomegranate ice cream I have tried, so I never mention it -- I just tell people my second favorite, which is not even a fruit flavor.
People will also take a cue on the range of acceptable options from the list -- for example from the list of five I mentioned, they may think that only single-flavor ice creams are being compared, so choose "coffee" because "mint chocolate chip" is a blend of two flavors. Similarly, I think many people initially describe themselves as "man" or "woman" because they felt like they had to pick the one that was closest, rather than define their own category. We choose from what we feel is the acceptable list of options and for many people that list is extremely short.
I am confident that if we stopped asking binary questions or asking people to choose from a short list, we'd find a much greater variety in the ways people identify, and a greater number that identify outside of the binary.
*I have loads of non-binary, neurodivergent, and artist/writer friends so I know this isn't true for most people who read me, but most people in the general U S population will not make their own line and write in their own choice!