From my very limited experience, it seems fairly common for soldiers to identify their arms by carving initials, perhaps unit names or other identifying information on the stock. I visualize a bored soldier sitting around a campfire occupying his time by making his weapon his own.
Brass plates, etc, seem to have been added later. I visualize a veteran or his family member applying these to record their history for future generations.
One non-CW example I am unsure of is British service muskets marked with numbered brass plates, apparently to identify the unit to which they were issued. Perhaps some US units followed this practice as well.