In the mathematical literature there are examples of graphs where the vertices form a group - the most famous example are probably the Cayley graphs.
I'm curious about a somewhat dual situation. Are there examples in mathematics of multigraphs (many possible edges between the same two vertices), where the set of edges between two vertices forms a group in an interesting way?
I'd be particularly interested in the case of an abelian group, that could be interpreted as "the sum of two edges is again an edge".
While this is well outside my area and so I wouldn't be fully comfortable to say "no", I think this extended comment may be of some interest.
The idea of attaching group elements to vertices is "in the spirit" of other mathematical ideas, in a way that's harder to interpret for edges. The Cayley graph originates in geometric group theory, which aims to study a group by considering spaces on which the group acts. The rough* idea of the Cayley graph is to turn the group itself into a space, since it already has an action. Well, at minimum, such a space must have group elements as its points! And it is reasonable to think about vertices of a graph as points of a space.
When you flip it around and ask "What are the edges of a graph doing?" the answer is that they are describing the existence of "a relationship" between two vertices. For the Cayley graph this means that two vertices differ by a generator, but of course there are all sorts of other relationships that may be possible.
Critically, though: note that this answer is really about the set of edges between two vertices. The set of edges is much less coherent as a "uniform" collection. There is a reason that when a group is represented as a category, then yes the group elements become edges (arrows), but there is only one vertex (object)!** Doing this "uniformizes" the edge set by forcing all edges to have the same endpoints.
The way I see it, what a group really "wants" to do as a collection of edges is not to uniquely label every edge of a graph, but rather to label the collection of edges that originate at a particular vertex. After all, the whole point of a group is to act on something (usually by symmetries); so it is natural to ask what the action does. This gives rise immediately to a graph structure (with edges $x\to gx$), and edges are labelled by group elements, but of course if there is more than one vertex then each element appears many times.
(* "Rough" in the sense that literally speaking, this is at odds with Lee Mosher's interpretation, and theirs is better. The distinction that Mosher is making is important: if you start with a graph that is unlabelled but I tell you it is a Cayley graph $\Gamma(G,S)$, you will not be able to find the identity element, even if you know $G$ and $S$, and even if the edges are already $S$-labelled. So we are not literally making $G$ into a space, but only making a space that "looks like" $G$.)
(** But, you may also be interested in groupoids.... It's not an answer to your question since it really is a cheat; groupoids are more or less defined to be the edge sets of graphs. Well, that's not quite true, but it's not far off.)