Is it normal to be asked for your passport number when reserving apartments in Italy?
Yes. The apartment owner is required to report who is staying in the apartment. Nothing to be concerned about. (And what nefarious thing would they do with that info anyway?) In fact, if they do NOT ask for your passport number, it could be that they are engaging in the popular Italian sport of tax evasion and will not be providing info to the authorities about their guest.
Michael is correct. Common practice and legally required in Italy.
Great-thanks for clarifying.