1968 was the mandate from my quick search.
Edit: yep Gun Control Act of 1968
https://lawcenter.giffords.org/federal-law-on-dealer-regulations/
Manufacturers still did for record keeping, especially the house hold names but I'm not sure others would have. For example, I have a Winchester Model 70 from 1955 that I looked up by the serial number.
H&R put the serial number under the grips on some old models.