The “registeredness” of a card is indeed stored on the physical card. Was never really a problem when there were tube ticket offices as you would just complete the form and register it there and then.
A card can still, in theory, be registered at an Oyster Ticket Stop. And presenting a card that thinks it is unregistered on the staff ticket machine menu under Discounts and concessions will prompt entry of a postcode and secret word, which will also serve to set the “registered” flag on the card.
IIRC, there are basically 2 separate systems that recognise an Oyster being 'registered' - and they don't speak to each other! One is the online, self service, registration and the other is the system that allows the station ticket vending machines (TVMs) to allow discounts to be added.
I recall asking people to fill in the forms required to add (in this case, PRIVs) to their Oysters. I then had to re-enter the secure suite (behind the TVMs) and input those details on the TOM/SAF (don't recall what that stands for, but station computer) and then have the passenger
write down their password on a piece of paper that I then inputted for them as they obviously weren't allowed into the secure suite! That meant I had the names, addresses, DOB, and passwords (in this case it was a family, so included the NR staff member's children too). This was circa 2015 I guess. The paperwork was a dodgy system even then.
So there was a weird 'double registration' system even then. Later on, I would
fake the registration on the TVM by inputting the station postcode instead of all that TOM/SAF nonsense.
Edit: When did ticket offices close on LU? Could have been early 2016 when this happened.