There isn't really a generic Buddhist notion of "school" or seminary, because the way most tendencies of Buddhism are propagated is that students are taught directly by monks and are frequently known primarily by their lineage (as in "x was a student of y who was a student of z" going all the way back to Bodhidharma or even the historical Buddha). It's equally unlikely, though, that a school that produced priests for Buddhist temples would be too overtly Shinto, although for sure, it's relatively rare that there wouldn't be some overlap between Buddhism, Shintoism, and "New Religions" (like Soka Gakkai) in a setting like that, because all of those belief systems are syncretic and much less uptight about mixing than, say, Christianity. Most likely a religious school would be affiliated with one of the bigger, more popular tendencies like Nichiren or Pure Land and would exist primarily to train the next generation of administrators for the larger organizations (represented by the individual temples and monasteries). So Heidi would have been going to one of these to learn how to take over the family business more than to get instruction in Buddhist doctrine.
Ultimately the difference between Buddhism, Shinto, and new religions are distinctions that would be pretty mysterious to the vast majority of Japanese people, even those who profess some kind of Buddhist or Shinto faith. Buddhism isn't really a religion in the same way that Western faiths like Judaism, Christianity, or Islam are (the idea of classifying followers of the Buddha under an umbrella called "Buddhism" is something that was done by Christian missionaries, in fact – until that point there were just monks, abbots and nuns that were part of a lineage / tendency, typically associated with a specific monastery or temple – this is a vast oversimplification, but maybe enough info to be interesting).
Sorry for the rabbit hole; as a semi-Buddhist myself and somebody who's spent a long time teasing apart how this stuff works in practice, I find it really interesting and fun to talk about.
