On the one hand, this one is hardly unique to Mighty Mouse. Cats are almost always cast as the villains in the "cat vs. mouse" cartoon conflict, even when, as in the case of Tom and Jerry, the mice are often at least as assholish as the cats.
On the other hand, even when cats go up against dogs, who have as many advantages over cats as cats have over mice, the cats still aren't portrayed as heroic by contrast, which means that this has less to do with childhood cartoons just naturally favoring the underdog (heh), and more to do with a bit of a bias against cats in particular (and I say this as someone who doesn't even LIKE cats).
It's interesting that certain traits get ascribed to certain animals - in the German language, in which all nouns possess genders (even inanimate objects), dogs are always male and cats are always female, regardless of the actual gender of the specific animal in question. What is it about cats that is culturally coded as "female," then, and could it be related to the reasons why cats are so often cast as bad guys, even when it contradicts the internal logic of the stories themselves?
Somebody else will have to answer that, because I've just typed way too many words on this already.