I think a part of it is a slight weirdness of Haskell typeclasses. In math, the noun "monad" refers to is a triple containing a functor and two natural transformations. The equivalent noun in Haskell would be the typeclass
instance for a particular type. However, because Haskell (basically) only allows one instance per type, you select different instances by choosing a different (possibly equivalent, as with newtype) type, and so there is some natural conflation that arises in the language people use to talk about these things.
The dynamic may be slightly clearer for semigroups. In mathematics, a semigroup is a pair containing a set and an (associative) operation on that set. In Haskell we say things like "ByteString is a semigroup" but that's short for "there is a semigroup where the set is all possible ByteStrings" and you assume the people you're talking to understand what the operation is because there is only one instance of Semigroup commonly defined for ByteString.