Humans are not an invasive species. We migrated to everywhere we currently live. Invasive species are usually defined something along the lines of being a species that was introduced somewhere (by people) where it didn’t previously exist and is harmful to its new environment. We meet the harmfulness part of the definition but not the introduction part of it.
There are native mice all over the place. Yes, some are introduced/invasive, but there are also plenty of native ones too. If you live in the Americas, here’s the subfamilies they make up:
https://en.m.wikipedia.org/wiki/New_World_rats_and_mice
Mice serve as food for animals like owls, hawks, falcons, snakes, skunks, etc. Cats killing these animals’ prey makes it harder for them to find food.