I don't think it's nonsensical, it's just another name for the same thing. E.g. in the Haskell wiki it says, "the Error monad, also called the Exception monad".
I was unaware of "Exception monad" being an industry equivalent term for Either/Error. Given no other context, desiring an "Exception monad" could be interpreted as "a type which handles raising Exception types." Which is how I did.
Thank you for clarifying this for me.