In the United States, how do you get car insurance (auto liability) which is valid no matter what car you are driving?

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.

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".

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.

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.

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.