I suspect the US would not have many, due to perception and lack of access to decent courses.
in the US and Aust, we would just call them social groups and they would be resigned to public tracks, resort or lower end private clubs. But the UK, some of them are quite well regarded. The Lucifer Society for example is very exclusive with membership being extremely sought after.
And of course they do at times play on the best tracks. I was a member of the London Stock Exchange Golf Society and we played on Sandwich, Deal, Walton Heath etc, as well as the Carlton Club Golf Society and we had annual matches on Rye, Swinley, Woking, NZ etc
This regard stems from their age and membership. In many cases, the membership is made up of the great and the good, and they are very old at times. The Royal Perth Golfing Society dates from 1824 and was the first ever club to be accorded the Royal Title in the UK - it has a clubhouse but has never had a course.
This would be impossible to replicate in the modern age without a course attached