What is the edge set of a multigraph?

There are different ways to do that and it really depends on what you want to do.

A very nice way is to define an undirected multigraph as $G=(V,E,\text{sig}: E\to \mathcal{P}_2(V))$. Here, $V$ is just any set, we call it vertices, $E$ is just any set, let us call them edges and $\text{sig}: E\to \mathcal{P}_2(V)$ is a map that tells us for a given edge $e\in E$, which two vertices $e$ connects.

Why it is superior to the multiset-option?

Alternatively, you can define $E$ as a multiset, i.e. $E \subseteq \mathcal{P}_2(V)$ together with a multiplicity-map $m: E\to \mathcal{N}$.

I prefer the first way with this signature map $\text{sig}$ over the multiset variant, because it matches the intuition of "different" edges better. IMO the multiset-option describes edges with multiplicities but not multiple edges.

Assume you want to express something like "Take two different edges $e_1$ and $e_2$" formally.

For the $\text{sig}$-option you say "Take $e_1,e_2 \in E$ with $e_1 \not = e_2$". But for the multiset-option, it gets difficult because just saying "Take $e_1,e_2 \in E$, $e_1\not=e_2$" is not enough!

I believe the term multiset is used to refer to a set that may have duplicate elements. This term makes sense here, especially since it is cohesive with the term multigraph. If you really want to keep the edgeset as a set you could let each element of the egdeset be a pair that consists of the edge itself and the mutliplicity of the edge. So the edgeset of the multigraph you posted would be $$ \{(\{1,2\},1),(\{1,3\},1),(\{2,4\},3)\} $$ If we go with the multiset term, the question then becomes this: should we call it a multiedgeset or an edgemultiset?