Do not be so sure about that. Manufacturers started putting serial numbers on their products before the US Civil War. I believe it was for their own purposes, most likely inventory control.
I would also consider it very likely that even if some of the various guns both men used over their lives had serial numbers many had no more than a manufacture's mark (brand symbol).
Well, I wasn’t sure, but made the smart remark anyway.
“... Manufacturers started putting serial numbers on their products before the US Civil War. I believe it was for their own purposes, most likely inventory control. ...”
There was little point to serial numbers before the 1840s, except as Fraxinus mentioned above. Best guess is that inventory control might be needed for tax purposes.
Each gun was so unique that the parts would not interchange. Not much point to serializing items so unique they could not be swapped around. Repairs to military small arms always required individual fitting of parts.
The US War Dept strove for years to produce small arms with interchangeable parts, hiring inventor John Hall to be the Master Armorer at Harpers Ferry to make it happen. Hall spent the time and public funds developing his breechloading rifle and conning the War Dept into adopting it.
The first US small arm with completely interchangeable parts was the Model 1842 musket in 69 caliber. It was also the first US issue small arm to be designed with percussion ignition (all earlier one had originally been flintlock; may were later converted to percussion), and the last primary issue shoulder arm not to have rifling (many were rifled later).
The War Dept did not bother stamping serial numbers on small arms until the adoption of the M1873 rifle the “Trapdoor.” Accountability and the threat of theft from armories may have been the motivators: labor unrest, socialism, and anarchism intruded on the public consciousness in the 1870s.
For many decades, commercial gunmakers hid serial numbers under the grip panels or stock tangs. Not on every model: Colt’s numbers appeared on the exterior surface, and the Winchester M1873 bears its number on the tang - a separate part from the receiver. Curiously, budget gunmakers like H&R hid numbers under the grip panels by stamping them on the gripstrap. So did Remington, on their M95 Double Derringer, which enjoyed a long production run from the 1860s until 1935.