Real Japanese police cases/wallets have both the actual metal badge and a blue ID card which is written in both English and Japanese, with the officer's photograph, name, ID number and rank.

If your entry stamp etc looks legit (valid, non-expired), that's what they were all probably looking for. You would of been in more trouble with no passport on you, or an expired stamp, that usually means going down to the koban/station and lots of extra paperwork. The police tend to not bother much after finding out everything is legit. If anything was amiss there would of been more questioning.