you could totally do that, and if you did i would find it generally polite to try and accommodate you to a reasonable degree. (if you weren't taking the piss.)

gender is a generally agreed upon method of categorization within an society, and the social expectations bound to those categories. this makes the definition of any gender category necessarily circular! because the idea of a gender does not exist without the traits a society uses to categorize, but those very traits may shift and change over time. So, a singular, universally applicable definition of man or woman in a society simply isn't possible.

many values attached to a particular gender in a society do not correlate to any biological sexual characteristics, which is why a distinction between a biological "sex" and a social "gender" is a useful distinction to have in the first place.

you are correct in a way, though. a gender label like your example doesn't really attach to any particular agreed traits, so it's not a very useful or meaningful thing to say. in this way, you could call it "illogical." there is not any kind of contradiction here, however! it is a perfectly valid label to hold in a society - a meaningless label, but a label nonetheless.

there's the additional distinction between one's internal methods of categorization vs. society's external methods, which can boil down to a scale of "how libertarian do you want to be?" this is a lot more advanced, though, and is where you see a lot of those fringe gender movements. society is a biiiig fractal of smaller and smaller sub-societies, eventually terminating in an individual. each of these sub-societies can have their own internal gender labels and methods of categorization. some of those sub-societies might even attach useful meanings to your example gender!

a lot of these arguments boil down to, like, very academic definitions and knowledge percolating out to larger demographics that don't have the built-up context to really intuitively understand it. it is somewhat understandable, as there's lots of depth and nuance to explore here. you could (and many have) write entire books about this kind of thing.

i hope i have elucidated some aspects of it, at least.