Question: Almost all auto insurers I speak with require that I give them the model and make of the car I will be driving. For example:

* Chevrolet Silverado
* Honda Civic
* Subaru Outback

Consider if I got in a car accident when I was...

* driving a friend's car
* If I was on vacation driving a rental car
* if I was driving family-member's car when an auto accident occurs

Any insurance claim I file would be denied, because I was not driving a vehicle covered by my policy.

How do I get auto liability insurance in the United States to cover ***ANY*** car I am driving?

Answer: 
Auto liability coverage follows the driver, no matter whose vehicle is being operated, so long as the operator has the owner's permission to use the vehicle.

You will see the opposite stated in many places. But that is an answer to a different question. For example, say you are driving a friend's car and get into an accident. State law may require your friend's policy to cover you. In that case, you have no liability because your friend's insurance covers the accident. So the question of whether or not your liability insurance would have covered you if you did have liability becomes a purely academic one.

If you have liability, you have coverage. Someone else's insurance will cover it in most cases, so you won't have liability anyway.
Double check with your insurance company's agent but ...

In the case of an accident, the car owner's insurance will pay first. The drivers insurance pays second. Points (a system used by most, if not all, US states to establish who are bad drivers) go to the record of the driver.

I learned this 20 years ago when I was going to loan a car to my English cousin. Interestingly, we presumed that points would not cross the Atlantic.

You might want to discuss with the owner who pays the deductible.

There are also limits on what an insurance policy will pay. Most policies will not pay enough to cover a wrecked Ferrari. So, borrowing a Ferrari is not likely to happen.

Rental car companies charge absurd rates. Figure the administrative costs and they might be renting to drunk drivers. A US company is not likely want to cover a rental in Mexico or Europe, or likely anywhere but the 50 states. However, AAA has reasonable insurance for that.

PS: They want to know your daily driver since that is what counts for other coverages, et cetera.