Why is it a mosquito *bite*?
We use bite for insects (or other beasts) that harm us with their mouth parts, such as snakes or spiders. We use sting when they harm us with a defensive device (i.e. sting), like in the case of wasps or bees.
Mouth parts do not necessarily need to be mandibles that open and close. They can have different structures and ways of working.
Female mosquitoes do have mouth parts with which they bite us. Look here. Therefore, and according to the previous logic, the phrase "mosquito bite" is correct.