Typically, insurance "follows the car" rather than the driver. An individual that does not own a car can generally obtain "non-owners" coverage that provides liability coverage to him/her regardless of the car driven.
Comprehensive coverage may be the closest to the kind of coverage about which you ask. It provides physical damage coverage to the vehicle against risks unconnected with collisions (such as a tree falling upon it, a rock hitting the windshield, and various other risks outlined in the coverage). However, comprehensive coverage is generally sold and purchased in connection with collision coverage.
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