At least in the places I've been in the US, law enforcement doesn't seem to care if you have your physical license with you. They just look it up, since they are doing that anyway...whether you have it or not. I'm sure that varies, though.
(It makes no sence because they have a digital picture of you provided by DMV in their cruisers, along with proof of insurance?)
Driving is without a license in your "immediate possession" is definitely against the law in California[1] and I've never met someone that intentionally leaves it at home. Restaurants that card for alcohol also makes this a no-go for most people.
1: https://leginfo.legislature.ca.gov/faces/codes_displaySectio...
Of course, in the UK car registration plates are issued and attached by the dealer, before a car's first sale, and they never expire. So they can find the registered keeper of a vehicle just by looking at the plates.
Yet another situation where foreign registered vehicles are essentially immune from UK traffic enforcement.