Whatever info you gave when getting the card could probably be assumed to be old anyway. So you would also have to always use cash as, once you use a credit card, it's probably associated with that loyalty card along with the real information that can be assumed to be good.
Really? Because I use a loyalty card with my real phone number (well, I don't actually use the card, just the phone number; I never carry the darn cards), which -- because of a key collision in the stores database, actually gets someone else's card that was registered with the same number -- and sometimes use cash and sometimes use a credit card (which is mine, and has my name). I know that the person who actually registered the card is using it, too (or at least, someone else who is not me) from the information about rewards credits that gets printed on receipts (I never use the rewards credits, but its clear that someone else is earning them, at least.)
(Actually, to make this more interesting, in fact, at the same store, I do this with more than one phone number of mine, all of which have collisions which result in some other card being associated with the sales.)
So, tell me, how is the store going to build an accurate profile of any real customer using the data they are gathering from any of these cards?