This is a tricky question! A run is different than a hole because it "runs" in a line through the fabric. Stockings can have a hole, but it doesn't become a "run" until it begins to turn into a line moving away from the hole. Other fabrics could potentially have "runs," but I don't think it's common to describe them as such. You would just say that your dress has a hole or your sweater has a hole that is beginning to run. Saying you have a run in your dress or a run in your sweater would sound a bit strange, unless it clearly looked like a run you would see in a stocking. Maybe others will help answer your question for a different perspective.

I would say with most confidence the answer is no. Run in the stocking phrase is only used for this particular issue. Why, , I have no idea, but for every other damaged material we never use run. Use tear, rip, tore, hole, etc.

It has to do with the way the material is made. It is a 'run' if the broken thread causes the thread to disconnect from the threads also above (or below) it, and so it continues above that, then above, till a long line of disconnected threads is formed. This line looks lighter in colour, and it has only horizontal threads because it loses all its vertical threads. This sometimes can happen also in other materials, like in some knitted sweaters. You have to sew that hole before it runs into a line along your sleeve.