The multi-set monad and modules

Question

I am trying to analyze the category of algebras for the finite free commutative monoid monad, aka the finite multiset monad. This monad is frequently described as having a multiplication that takes a formal sum of formal sums to a formal sum. We think of the multiset $\{a,a,b\}$, on set $A =\{a,b\}$, as having "2 $a$'s". Thus, we seem to get (for free?) a notion of multiplication by Natural numbers and also a notion of being able to add elements $a,a$ to get $2a$. I really don't understand why this is, but I am trying to stretch this analogy as far as it will go. To that end, I would like to see the category of algebras for the multiset monad as somehow equivalent to some category of modules. Can anyone give a little description of why this intuition may be reasonable, or unreasonable? I have even heard that there is a form of the multiset monad that is a vector space monad, whose category of algebras is, indeed, a category of modules. We see this here in Jacob's 2013 paper. In this paper, Jacobs states that the category of algebras for the multiset monad, on a semiring $S$, is the category of $S$-modules. I am guessing that the finite multiset monad has its category of algebras given by the $\mathbb{N}$-modules, where $\mathbb{N}$ is the semi-ring of natural numbers.