Insurance can cover a vehicle and/or a person. For example if you own a vehicle it will be listed on your policy and insurance would follow the vehicle for when you loan your car to others. Or if you do not own a vehicle you can have non-owners auto insurance in which case that insurance would cover you, but normally as secondary insurance on a vehicle. The insurance of the owner of the car, the one you borrowed since as a non-owner you would not have one of your own, would be primary.
State laws will differ but most require the car you are driving to be insured by the registered owner. If you are driving a car that the owner has not insured and thus does not have primary insurance for it, then your insurance as a driver may not cover an accident. The driver's insurance is normally secondary and if there is not primary insurance on the car the driver's insurer may be able to deny a claim.
Also if you are driving in a car that does not have insurance and are stopped by the police you would have no insurance card to show law enforcement since the driver's own insurance card would not show that specific car on their insurance card for their own vehicle or non-owners policy.
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