You can throw away your supermarket loyalty card, and stop getting those 2% discounts. You're not powerless. You're making a choice.
The only thing that is guaranteed is that I've lost the discount. Recording accounts from credit, debit, or check payments leaves the customer equally screwed.
More likely: I request loyalty cards under assumed names and discard or exchange them often.
Though that likely leads to other data trails. Anonymisation and tradecraft are hard.