Let G be a simple graph and let us color its edges so that the multisets of colors around each vertex are distinct. The smallest number of colors for which such a coloring exists is called the irregular coloring number of G and is denoted by c(G). We determine the exact value of the irregular coloring number for almost all 2-regular graphs. The results obtained provide new examples demonstrating that a conjecture by Burris is false. As another consequence, we also determine the value of a graph invariant called the point distinguishing index (where sets, instead of multisets, are required to be distinct) for the same family of graphs.