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.

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][0] 


[0]:https://travel.stackexchange.com/questions/147247/us-passport-card-for-identification-not-customs-purposes