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][1] 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][0] 


[0]:https://travel.stackexchange.com/questions/147247/us-passport-card-for-identification-not-customs-purposes
[1]:https://en.wikipedia.org/wiki/German_Chancellery