I think this has to do with the rental car lots being so close to the terminal building at BNA (basically adjacent to the main building and at the base of the parking structure)...the authorities are paranoid that someone will stuff a bomb in a trunk and detonate it right in the rental lot. I've never been asked for ID, but I have been asked to let them check the trunk. BTW, the "big boys" (Hertz/Avis/National) are under the parking structure and don't check trunks in general, it's the lower-end companies (Thrifty, Dollar) that have lots closer to the terminal and have the checkpoint at the return entrance (at least that was the situation last time I was there).
I've always liked BNA's rental-car setup for its lack of an offsite facility and those damned shuttle buses--I hope this doesn't change.