In Carlo Beenakker's answer to this recent MO question, it turns out that the name "monoid" was first used in mathematics by Arthur Cayley for a surface of order $𝑛$ which has a multiple point of order $𝑛−1$.

On the other hand, a (old) name for the blow-up is "monoidal transformation". I suspect that there should be a link relating these two classical terminologies, but I was not able to find one. So let me ask the following

Question. Why was the blow-up classically known as "monoidal transformation"? Was this related to Cayley's "monoid"?