I didn’t even think about the German word, which is Bierdeckel - literally beer cover. Well the boring word for it would be Untersetzer, which is the substantive of “to place underneath”.

Additionally to what else was said, I’ve seen them used in beergardens to cover the drink to prevent bugs from drowning themselves in your drink. I guess then the coasters don’t serve to protect the table anymore.

Es gibt kein Entkommen!