Your credit card number being stolen is a problem for your bank, not a problem for you. You can't steal someone's identity with a credit card number.

The concern in this case is when there is some social problem with being in Mom & Pop Inc's customer database. There are probably some people that buy some things that they don't want other people to know about. When the database gets hacked and you are linked to being their customer, that is the unfortunate and potentially damaging information leak. A credit card just gets reissued and the bank reverses the transaction. No big deal.