This comedy routine is top notch, but seriously:
Anything I can do to the bag to avoid this problem? It doesn’t appear to be dirt.
I don’t know. If I had to guess, I assume your bag is made from crust leather. Some Italian companies use crust leather because
1) they can sell customers on the whole “artisanal” craftsmanship thing
2) they are often smaller workshops when compared to Chinese companies, and thus benefit from having some flexbility. If you have a huge stock of crust leather you can color them however you wish depending on the orders that come in. So instead of buy 100 hides in black and 200 hides in brown (and hoping you sell exactly this number), you can buy 300 hides in crust leather and color them however you need.
If I had to guess, I assume the color rubbing off onto your clothes are the coloring agents/ dyes that were used to color the bag. I assume you can either seal up the surface, which will probably look ugly. Or you can strip the color off, which will also look ugly.
But I don’t know for sure if that’s crust leather. Just my assumption from what you described.