They get so much more than that. If you use a credit card at the store, the record now knows about this card, and any other credit card you have ever used while using that rewards program. This can all then be connected to other databases that track personal credit card spending in similar ways. A surprisingly detailed profile is quite likely to be out there on all of us, even if we are not always identified by name. If you have last 4 digits of a card plus zip code... You have a whole lot.
My memory is a little vague on the sources for this but I will try to dig up some relevant stories.