[Nrsolis]

Correct.

Liability is the "required" part of insurance in most states. It follows you from car to car.

What has likely happened is that they person who rented the car didn't have insurance of their own. That seems likely, since it's kind of pointless to have automotive liability insurance if you don't own a car. Even more so if you're used to purchasing insurance at the point of sale when you rent a car.