They have at least two religions: Shinto and Buddhist.
I don't know how much following either of them gets today, but I do know that during the feudal period the two religions were associated very strongly with the upper castes (Buddhism) and the lower castes (Shinto).
I do know that religion is not an identity to the Japanese in the way Judaism is for Jews. I know people who simultaneously consider themselves Jewish and atheistic or agnostic. That seems bizarrely contradictory unless you understand that "Jewish" is both an ethnic identity and a religious identity, so even individuals who reject the religious part can still embrace the ethnic part.
Like the gp, I would be very interested to know of any other group that ties religion, ancestry, and culture together into a single unique identity the way Judaism does. I think that Persia under Zoroastrianism might have been like that, but the conversion to Islam ended that.
Interesting. I've to admit my ignorance again - no idea.
How 'religious' is Japan? I was baffled on the day I learned that you cannot marry without a ceremony with a Rabbi here if you're a Jew. In DE it's (all romance removed..) signing paperworks and optionally going to a church if you like that sort of thing. Christian ceremonies have no influence over your status though: You're married _iff_ you signed the paperwork, whether a priest held your hands or not. Here this seems to be different. And this is just one thing that confuses the hell out of this particular secular guy.
Don't want to bash something I don't understand. Consider this just a try to emphasize my confusion and venting some 'Boy are things different' steam.
This confuses (and infuriates) a lot of native Israelis too. This is more the result of pressure by orthodox interest groups and attempts to maintain status quo. Just another in a long and confusing list of factors shaping the Israeli and modern Jewish culture.