I have been a member of USAA since 1972 and whenever living in the UK have insured for auto, general liability, life and property with them. 95% of eligible members insure with USAA although not necessarily for all risks. USAA is often not the cheapest for buildings fire cover -- something at USAA Ltd. does not offer anyway.
If you carry motor coverage with USAA Ltd you get, free & gratis, cover for rental car usage in the USA. You also get green-card cover for the rest of Europe without any particular limitations. You get, as others have said, cover for any car you drive and any person who drives your car with permission. In the USA you get medical cover. For the UK, cover is somewhat less than you may be used to because of the EU Motor Insurance Directive which guarantees industry cover (through motor insurance bureaus) for any car carrying EU license plates from a different EU country, even if those plates are forged, stolen or otherwise void.
With USAA you will not be bothered by the conundrum of "no claims discount": some months ago a driver sideswiped my parked car. Fortunately a traffic warden was there. He left a note and when we spoke insisted on paying cash to the car body repair shop -- doubtless because of fear of losing that discount.
Many years ago I was driving on a French motorway when I had a tire blowout. Our estate car turned 180 degrees and (saving our lives) latched onto an Italian articulated lorry, whose driver saw what had happened and slowed down to a stop. As we were moving household goods to London the boot was full; the crash wrecked most of its contents.
(1) A wrecked untaxed car has to be removed from France, and USAA had it brought to the UK for scrapping, only to find it was untaxed there (I had been a diplomat). Nonetheless they paid me the full book value they had already quoted and said "never mind about the tax" once they understood the situation.
(2) A week later I had a call back from USAA: "You have property cover with us too. Did you lose any household goods in the crash." They then paid me new-for-old.
(3) I will never insure with anybody else but USAA, whatever it costs.