The concepts themselves aren't difficult to understand. It's just that some people go out of their way to use uncommon terminology that ends up obscuring what they're trying to say, rather than making it clearer.

Is there more common terminology for "a set with a binary operation that is associative and has an identity"?

Any name that we'd come up either wouldn't capture the general case or would be just as arbitrary as Monoid ( ignoring historical usage). The gist of this argument is that Pacabel and the like can't describe the concept using the traditional OOP evocative naming style but insist that term "Monoid" is wrong because of some irrational aversion to terms from abstract algebra. There's really nowhere for this argument to go...

Imagine trying trying to describe a monoidal category or topoi in the OOP style, it's like trying to describe what a Fourier transform is to someone who insists on using roman numerals.

> it's like trying to describe what a Fourier transform is to someone who insists on using roman numerals.

Or describing Latin declensions to someone who reads kanji. The key is that, if the audience admires Latin speakers, you can appear smart.