It was against regulations to mark-up or deface US Military issued gear, was it done, on rare occasions, but was not the norm. Rifle slings, underside of cartridge/cap box flaps, underside of haversack flap and very small initials in obscure areas, under the brim of a forage cap.
Many Confederates came into the service with their own weapon and thus the practice/regulation really never took hold, they marked and carved at will, even when better weapons were issued or captured.
Collectors have a feel for carved stocks and look for additional signs of the piece being Southern, burn out by the bolster, severe pitting around the nipple as a result of the heavy use of niter and sulpheric acid in the manufacture of percussion caps. The North had ready supplies of Mercury while the South had a very limited supply and heavily mixed fulminate, which upon detonation, was extremely corrosive to both wood and metal.
Many carving were post war, when a prized weapon was being put away for posterity or handed down, others obtained battle honors.
In terms of recognizing ones weapon, no two stocks are the same and are readily recognized by a soldier, who lives with it daily. Soldiers found unique ways to make the weapon theirs