Both barrels on a light target load won't put that much stress on the gun and stock, but tripping both barrels on heavy field loads, slugs, etc., can put a tremendous amount of stress on the stock and could, possibly, cause the stock to fracture at the wrist or crack throuh the inletting.
I had both barrels of a single-trigger 12 gauge fire simo due to a serious problem with the gun, and even with moderate target loads it wasn't particularly pleasant.