No, monad is more complicated than that. There is a specific method that it has to have, which does not make sense on all types, and can also have more than one

No, monad is more complicated than that. There is a specific method that it has to have, which does not make sense on all types, and can also have more than one sensible definition for a type. "You want to abstract over it" gets you to a "trait", not a "monad".