A multiset is a pair $(S, f)$ where $S$ is a (ZFC) set and $f$ a function $f:S \to \mathbb{N}$ which assigns to each $x \in S$ a natural number denoting its multiplicity.
This is fine, but we're still defining multisets in terms of standard set theory.
Has "pure" multi-set theory been studied before?
I wouldn't say it's been studied extensively, to my knowledge, but this paper approaches the subject by surveying the literature, defining a first-order theory, and proving relative consistency with $\mathsf{ZFC}$.