Also, the elements are still combined associatively. With closure and and identity properties, the set forms a monoid under this operation. "Monoidness" is not a property of operation (in a way that the language is usually used, at least in math).
For that matter, "combine monoidally" is a phrase that has 2 Google hits at the moment, which indicates to me that in real life, you still have to resort to less-shiny words to describe things.
There doesn't seems anything hugely objectionable to me in saying "Bars combine monoidally". But if you prefer, one can say "Bar forms a monoid under baz and quux".
It may not be obvious to you that it increases simplicity merely to combine two words into one but across an ecosystem of perhaps a thousand Monoid instances it indeed is.