The Italian law says the following (Art. 6 par. 3 of Uniform Immigration Code, promulgated by Law 286 of 25 July 1998)
The foreigner who, at the request of the officials and agents of public security, fails to comply, without justified reason, with the order to show the passport or other identification document and the residence permit or other document certifying the regular presence in the territory of the State is punished with imprisonment of up to one year and a fine of up to 2,000 euros.
So the police officer is required to check not just your identity but also if you have the entry stamp when you entered the Schengen area. That stamp is posed on one of the pages of your passport.
In practical terms, it is unlikely that an American tourist would be requested to exhibit a passport to a police officer, but it can happen, especially if you don't look "very American" (I will let you infer what that means) but could possibly be a person at a high risk of being an undocumented migrant or terrorist.
If you don't belong to the latter category, the possibility of getting into trouble for not having the real passport with you is low, but I wouldn't venture too far from the hotel without one (like going out of town on a day trip).
Pick pockets are not really interested in your passport. They just want your wallet and/or (especially) your electronic devices. So keep your passport separately from your wallet, purse, and (especially) your smartphone.