It seems that there are significant numbers of Jews, Anglicans and Shintoists that don't believe in the theology and the supernatural stuff, but still identify as members of the religion and follow the traditions

Culture is pretty strong. My girlfriend oscillates between Christianity and Paganism and is an active member of the local Church of England. They're currently trying to draft her as a volunteer for all sorts of things (the standard punishment for public display of sanity and competence). I'm a sceptical atheist and I'm on the fringes of but still pretty much somewhat part of said church community. Mind you, the C of E is hardly that unfriendly to atheists ... and Richard Dawkins still visits his and neither it nor he catch fire as he walks in ...

Or: Yes, religion is supposedly about the silly ideas and mad old books, but only works if it's a community, and these will often include people who expressly repudiate the silly ideas and mad old books.

"If the Church of England relied on Christians, it'd be sharing a room with the Flat Earth Society" - Shelley (TV show), quoted from memory.