Do I need to register with the police in Italy?

Your hotel does it for you, which is why they ask for your passport at check-in. Italian law requires every accommodation to report guests to the local police within 24 hours, so the desk will photograph or scan your passport and hand it straight back. It's routine, not a red flag. If you're staying in a private apartment or with friends rather than a registered property, technically the host is responsible for the same declaration — worth confirming they've done it, though enforcement on short tourist stays is essentially nonexistent.