Originally Posted by Cat Herder:
I also have never been asked to leave my license with a car dealership or apartment complex, only my car keys. Why would anyone drive willingly without their license on them? 
Generally they take a copy before you drive away. In case you steal their car, they can file a police report and know who you are.
We are required to take a copy of DL every time someone finances a vehicle in my line of work. I've never had someone question it.