This drink is a bit too sweet for me, but I make them and drink them anyway because I want to feel as cool as the Dude. The Big Lebowski is a cult classic and therefore this drink will always have a cult following. This drink is just the boozier version of a Kahlua and Cream.
Another variation on this is the Black Russian, which is just coffee liqueur and vodka. These drinks started showing up in the late 40s early 50s but it wasn’t until the film The Big Lebowski that the White Russian gained popularity.