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: 
Let's look at your examples one by one:

> 
> driving a friend's car
> 
> 
> 

* Your friend's insurance would likely cover you instead of your own

> 
> If I was on vacation driving a rental car
> 
> 
> 

Many, but not all, insurance policies cover cars you rent by the insured drive. You can also get insurance for a fee from the rental company. A 3rd option, is that some credit cards provide insurance if you rent a car using their card.

> 
> if I was driving family-member's car when an auto accident occurs
> 
> 
> 

Same story as if you were driving your friend's car. However, if you LIVE with that family member and aren't listed on their policy it may be an issue depending on how frequently you drive it.

This system makes sense because they price the insurance in terms of how much risk they are accepting. It would be much more expensive for them to pay a claim on a Ferrari than a Chevy Malibu, so they have to price it that way. Assuming you could convince an insurance company to cover ANY car you could possibly drive, they would have to assume the worst and charge you extremely high rates.
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.