I don't think anyone is going to say that Haskell's type classes in any way fit the traditional notion of a type. It's more the exception that proves the rule in this case.

I know it's a common saying, but I've never understood how exceptions can prove rules.

It is a very old saying from when "prove" meant what "test" means now. Exception is meant in the sense of exceptional, unusual. "The unusual event tests the heuristic."

It's actually a really simple thing that most people misunderstand (such as the parent). It means that if you only provide an exception, it implies a rule. For example, if you say:

Parking allowed 3pm-5pm

that implies that parking is not allowed at other times.

It's just something stupid that people say when they've been caught over-generalizing.

Pointing out how flexible Haskell's type system is just proves how inflexible type systems generally are. It doesn't really demonstrate that most type systems are useful.