What is implied by the handling of multisets by N-modules?

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Multisets are containers, also called bags. A multiset is a set that can have repeats:

$$\{ a, a, a, c, b, c \}$$

Usually when researchers talk about multisets, they use this kind of presentation:

$$ \psi = 3| a \rangle + 2| c \rangle + | b \rangle $$

In fact, $\psi$ is actual an element of an $\mathbb{N}$-module. $\mathbb{N}$-modules are the Eilenberg-Moore category of the multiset monad. Why do researchers always represent multisets as $\mathbb{N}$-modules? What is the deep implication here?