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: 
There is a kind of insurance like this, for people who don't own a car but nonetheless want to be covered for any car they drive, for example when renting a car or borrowing a friend's car. This kind of insurance goes under a couple of different names. When I first got it over 10 years ago, it was called "no-name insurance". But when I did a web search for that term just now what came up instead was "non-owner car insurance".
Ask insurers directly about "named non-owner" policies. They're uncommon, not all insurers offer them and those that do have to manually underwrite and manually quote the rate. It's not going to show up in any automatic tools.