Why multiplication isn't the monoid of number instead of summation since both operations are monoidal?

Question

In Mathematics, the monoid of numbers is summation, why it can't be multiplication since both operations are monoidal (they both are associative and binary, and have an identity value)

Answer 1

You are right, but in order to disambiguate, you can talk about the additive monoid of positive integers, that is, $(\mathbb{N}, +, 0)$ versus the multiplicative monoid of positive integers $(\mathbb{N}, \times, 1)$.

Answer to your comment. To start with, you can look at the Wikipedia entry Monoid.