English “mites” are not called mites at all in French, they’re called acariens. (And in English, the study of mites/acariens is called “acarology”!) Since those are arachnids, they’re very different from moths/mites which are insects, but the word “mite” has the same origin in both languages.
In 1765, the Encyclopedia described mites as having either six or eight legs, and talks a lot about what seems to be the flour mite Acarus siro (which was called mite du fromage at the time but is never called a mite in modern French), so it looks like the word used to have a broader meaning in French.
We have pantry moths and clothing moths, which I don’t think I’ve ever seen in person.
I’m pretty sure I’ve seen a clothing moth’s caterpillar once.
I had just put on a dress I really liked when, looking down, I made out what I thought was a bit of white fluff hidden in one of its folds. I rubbed it to get it off the dress, and inside was some kind of worm, moving slightly. I’m usually tolerant and curious of bugs, but that one really freaked me out; I killed it right away out of pure reflex and threw it as far away as I could.
So cute, it looks like it’s wearing a little mask.