A list is a monad

I think this adds more confusion than it removes.

A list is not a monad. A list is a data structure; a monad is more like a "trait" or "interface." So you can define a List type that "implements" the monad interface, but this is not an inherent property of lists themselves. That's the sense in which a list "is a" monad: the OOP sense.

Haskell's List monad provides a model for nondeterminism. But that certainly isn't the only way List could satisfy the monad interface! It was a deliberate choice -- a good choice, possibly the best choice, but a choice nonetheless.

Isn’t it the case that for a given functor (on Set) there can only be at most one Monad structure?

Nope. It's that there's only one lawful Functor instance. But Applicatives and Monads can be multiple - lists are the classic example (zip vs cross-product)