https://en.wikipedia.org/wiki/Quiver_(mathematics)

https://ncatlab.org/nlab/show/quiver

A quiver is simply just the data of a category, i.e. a "category" without any of the laws, namely identity and composition.

They're not isomorphic to DAGs since Quivers can have multiple edges between the same set of vertices, in other words, they'd be the DAG equivalent of multigraphs (directed multigraphs, if you will).

For example, in the category of Sets, vertices are sets and edges are functions between sets, so between e.g. N and N there will be infinitely many edges (all functions between natural numbers) with a particular distinguished identity edge that maps f(n) = n due to category laws. So if you turn the category of Sets to a quiver, you'll have infinitely many edges N -> N and one of them will happen to be the identity function `f(n) = n` but you "forgot" its "identity" relationship/law when you reduced the category to a quiver.