maybe state insurance laws are different where you are but here in Texas, "liability" insurance is on you, the driver, it only covers the other persons car, or property or person. It does not cover the car you are driving or your medical.
You must have liability insurance to legally be allowed to drive. With liability insurance you can legally drive any vehicle regardless of being named on a policy.
Comprehensive and collision coverage stays with the car because comp and collision only cover that car, not drivers, not other cars.
In Texas you can get liability insurance even if you do not own a car. Many employers are required to purchase this for their employees who do not own a vehicle yet must drive a company vehicle.