That assumes you have to put in valid data for that. In that case, you are just "Customer #12345" versus a person they can correlate against.
Discussion of this: https://www.creditcards.com/credit-card-news/retailer-data-t...
I mentioned it in another comment, but the combination of PCI compliance and the use of chip+pin make it difficult for retailors to use credit cards to track an individual, but the credit card company can.
However, if the retailer then goes back and purchases the data from the credit card company, I agree, they will know who "Customer #12345" is with regards to the person.