I've never visited Japan, so I might be totally off, but based on what I've seen in anime and manga, it seems to me these family restaurants are basically mostly serving Western themed food, aren't they? So, in Japan, they would be what you'd call an ethnic restaurant (since they aren't specialising in Japanese or even East Asian cuisine), whereas in the West, they would be just nondescript restaurants (since basic Western food is 100% ordinary food commonly found everywhere in the West). Even in the West, you still have more specialised Western restaurants, like Italian, Greek, French, Scandinavian, etc restaurants, of course.
But like I said, maybe I'm totally wrong since manga and anime don't always reflect the reality, surprise surprise.