Having travelled in practically all European countries very extensively, I must say that the only places anyone asked for my passport/ID were the hotel receptions and airport gate agents. So while it's technically compulsory to carry your passport with you in parctice you won't face any problems if you don't. Of course it's always a good idea to carry some identification, in case of accidents, emergencies, etc.

(Once I was med-evac'd from MOW and the passport was at the hotel, myself in intensive care and the hotel wouldn't let anyone into my room to pick up my stuff. No passport - no med-evac. Quite an impasse until the hotel mngr could finally be persuaded that it's OK for Sergey to come and pick up all my stuff)