Suppose we have a set $S$ with a single commutative binary operation $*$ and an identity element $i$. Is $(S, *, i)$ necessarily a monoid?
According to this answer, a monoid just requires an associative operation and an identity element, so if I'm not mistaken, the question can be reduced to whether $*$ is necessarily associative. According to this answer, the answer is no.
I'm new to category theory, so I wanted to know if my reasoning is correct.
Your line of thinking is correct: all that is missing from your assumptions to conclude if you have a monoid is to check whether or not the binary operation is associative. This is not true in general, as commutativity doesn't imply associativity, demonstrated many times over in the second of your links.