Does the current implementation of
IGChromaticIndex in IGraph/M work with multigraphs? I recall reading something to the effect of the contrary but I can’t find that source again.
Yes, edge colouring functions are intended to work with multigraphs. Do note that the current implementation is not quite as efficient as for simple graphs.
Mathematica itself has the annoying limitation that it is not possible to set separate edge properties for parallel edges. Therefore, it is not possible to visualize the edge colouring in the same straightforward way as for simple graphs. However, the colouring can be computed and the functions you mention are not affected by the problem. The visualization will, unfortunately, have to be done manually.