Nope. You are supposed to bring the notice DMV mails you or the registration when you have the emissions test done. But SS did that months before hiring JW. Re-registering a car online is very simple, so I find it doubtful that SS would have JW go to DMV, wait half a day for his number to be called to register it in person in Maryland. It would make sense that SS would have had his registration renewed around the time that he had the emissions test done, but it never showed up on the carfax. Some states don't report re-registrations, and it appears some don't report all of them very well, but we don't know if the Mosler was ever re-registered. If JW had done it within the weeks that he was working there, it would have shown up on the carfax by now, IF MD reports all re-registrations to Carfax.
You definitely don't need your registration for repairs. If a dealership or mechanic reports repairs to Carfax, they use the VIN. I agree, the momento idea doesn't make sense to me either, but it makes more sense than needing it to maintain the cars.