alt Hacker NewsIt's not the worst name for the concept when you include "a male" and "a female" as prominent nouns in that noun class. If you adjust your language depending on whether you are addressing a man or a woman (or speaking about a man or woman), then it's definitely also social gender (as well as grammatical gender), even if those two concepts are separate.
Except there's no mandate that "a male" and "a female" are of different noun classes, nor are the nouns for man/woman abnormally privileged in most cases. I know Dutch has fused masculine/feminine nouns into a "common" gender, leaving the language with effectively only the common and neuter genders. If I remember correctly, a similar thing has happened in Swedish and Danish. Some languages have various concepts of animacy driving the system. Some languages have shitloads of noun classes.
You can adjust your language depending on the biological gender of who you're addressing in English, but English doesn't have grammatical gender in any meaningful way. The concepts are largely orthogonal.
Calling it gender really is just a bad, misleading name in the grand scheme of things.