From the viewpoint of the victims of a stereotype, I doubt that the degree of factual foundation which the stereotype possesses makes much of a difference.
Or based on a misunderstanding, but still coming from a real observation. For example in the US, apparently there used to be a stereotype that polish people are stupid, which came from language differences - polish immigrants had a lot more trouble with English than everyone else, or even were less likely to know any English.
But the general stereotype is not “completely made up” so to speak, it’s an observed average.
Then of course some total crap is always thrown in the mix but still.