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.
edit: If they are allowed to track based on credit card, this would be a good reason to use Apple Pay, or some other payment method that generates a unique "card number" for every purchase.
Edit: so you just want to hand over your info to Apple instead of some other big corporation?
[0]: http://www.slate.com/blogs/future_tense/2017/08/01/google_tr...
You might as well ask for web servers to not see the IP address of people connecting to it.
https://www.forbes.com/sites/kashmirhill/2012/02/16/how-targ...
"As [Target's] computers crawled through the data, he was able to identify about 25 products that, when analyzed together, allowed him to assign each shopper a “pregnancy prediction” score. More important, he could also estimate her due date to within a small window, so Target could send coupons timed to very specific stages of her pregnancy"
Read the entire article for a more horrifying anecdote.
Target may have been too successful in this goal; as Duhigg writes, one teenage customer in Minneapolis was "outed" as being pregnant by a coupon mailer sent to her house. Her father was justifiably upset to see Target offering his underage daughter discounts on diapers and cribs, though he was perhaps more upset when he discovered Target knew more about his daughter's personal life than he did.
Credit card companies also do this, though I believe in the EU and even US there are stronger regulations around what exactly they can send (vs. rewards cards).