It isn’t for all repeated vowels; only for when the 2 vowels don’t make a single sound. So “chicken coop” wouldn’t have a dieresis

It would if the chickens formed a business structure that was owned and democratically controlled by its member-owners.

That, is likely co-op.

That's the joke.

Unless it was a chicken coöp... One of few cases it actually resolves an ambiguity!