I grew up in the U.S. and we've always said that mosquitoes bite. It's true there are no teeth, but we think of a "stinger" as a separate body part, like the tail-tip of a hornet or scorpion. Especially since the mosquito is also feeding on us, we would definitely think of that as a "mouth" action -- hence "biting."
It does make it hard to translate, though, if I'm thinking of one action but the correct expression in another language is not the same action! This seems to happen especially often with Japanese. Thanks for explaining the reason behind the expression.