Yes, there was such a law back then - and it didn't have diddle-squat to do with product safety recall notification. It had to do with the fact that microwave ovens were in their infancy and posed a "potential" threat to be an incidental source of electromagnetic radiation interference to protected communication services (military and civilian radar; aircraft, public service such as fire police, private, citizens band, and amateur radio; broadcast TV and radio reception). By having your "nuker" registered with the FCC - if they received a complaint from someone about EMI (interference) they could narrow down the culprit with a minimal amount of effort and in a timely manner. Of course, back then, most people received their TV and radio signals via antennas - not cable which is much less prone to EMI.
Also, about that time, they came out with something called Lithium-Ion batteries. One of the first civilian uses was for cardiac pacemakers. These "nuclear powdered" pacemakers required registration with both the FDA and the Nuclear Regulatory Commission (or, whatever it was called back then). The main thing I remember is that there were about 20-pages of rules to read/comply with and about 10-pages to fill out for each patient.
If the microwave manual has a reference to a CFR - that will identify the law that was in effect at that time.
"It ain't what you don't know that gets you in trouble. It's what you know for sure that just ain't so." - Mark Twain