blaze-html is an example of something implementing the Monad typeclass just to get desugaring. Somebody wrote Lucid for exactly the reason you describe(you can't use things like forM_ with blaze because it blows up).

In any case, you might argue it isn't a mathematical monad, but it can still be a Haskell monad as long as it implements the typeclass.

I agree it makes the most sense if it implements the laws: Otherwise, it's like calling something a Stack<String> when it really deletes your home directory whenever you peek an element. Technically anything that implements the Stack<> interface is a Stack in the language, but it's not an abstract Stack.