For a car rental, proof of nationality and legal status, would be needed to prove that your driver's licence may be used, since foreign residents, after a specific time, must exchange their driver's licence for a local one. The same rule applies in the US when moving from state to state.
As a visitor under the 90/180 days rule (entry stamp in passport) it is clear for the car rental (and any policeman stopping you while driving) that there is no problem from that aspect.
A driver's licence (or US Passport Card) is not sufficient to prove your nationality and legal status inside the Schengen Area (lack of entry stamp) by any authorized person (such as a policeman). Keeping a passport in the hotel safe is a bad idea, because if you need it you won't have it.
It also depends on the reason they are checking. While visiting the Bundeskanzleramt this summer, my friend forgot his passport, but they let him in with his Kentucky Driver's License, since they were only checking against the nasty person's list.
Identity checks for age by non authorized persons the Passport Card or driver's licence should be sufficient.
This was also discussed here: US Passport Card for Identification (not customs) purpose