A lot of languages assign nouns to a noun class. They are (usually) not ascribing a biological gender to an object. "Gender" is a horrendously bad name for the concept.
It'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.
French and Spanish literally use a contraction of the Latin for he and she for different objects though. This is also done in Swedish.
Hon slår tolv. 'She strikes twelve', referring to the clock currently striking.
Grammatical use came first by far.
"Gender" referred only to grammar before it gained its modern meaning. The modern meaning was introduced in the 1950s/60s to differentiate social aspects (gender) from biological (sex). Of course people then started using it to just mean "sex", but if you use social definition I don't think it's a bad name for the concept.