Can you count the Cornetto films as a trilogy in the same sense as Star Wars 4-6? I love all three, especially the first two, but while they share actors, themes (to a certain degree), and pay homage to each other with some gags, there's no shared story or even shared world.
To me, it feels weird to put them in the same category as the Godfather or LotR films, etc. They're more like Hitchcock's "trilogy" of Rear Window, Vertigo, and The Man Who Knew Too Much. Like, trilogy broadly meaning "a complementary set of three things." I feel like once we open the category that wide, there are a lot of other outstanding sets of films that I need to consider.
Like, what about Robert Altman's Nashville, The Player, and Short Cuts? Or Scorcese's "greed" and "God" sets of films? Etc.
I suppose I am thinking of trilogy in the sense of a series of directly connected films - sequels.