There's a "really close correspondence" between quivers and categories, where quivers are directed graphs that can have multiple arrows from one vertex to another one and also loop arrows, which are arrows from a vertex to itself.  Isomorphisms become undirected edges.  This is a really good and precise way to think about it, because this viewpoint generalizes very nicely to some models of higher catgory theory, specifically A. Joyal's theory of quasicategories.