Yes, the case or primer would be microstamped, mostly likely by the firing pin.
One very serious issue with this is that it will preclude being able to replace a broken firing pin. A replacement firing pin could take months to procure, at who knows what cost, while a special replacement pin is made with the same number.
And not only would the cost of the gun go up to pay for the technology, but this is a patented process, so there are royalty fees and patent rights. In fact, this could be used to drive small gun makers totally out of business.
And, what is to prevent someone from obliterating their firing pin marking? If it is metal it can be polished away.
And it doesn't take a rocket scientist to see that in order to prevent this scheme from being thwarted that at some point firing pins may have to be altered so that replacement firing pins already in circulation cannot be used on the new guns.
And the frosting on the cake will the gunowners in the states adopting this will find themselves restricted to only those firearms meeting the new criteria.
At least Remington is finally getting politically motivated. All too often gun makers have either sat back and done nothing, or in some cases aided and abetted the anti-gunners through stupidity, greed or appeasement.