The phrase you want is 'loyalty card'. I have never heard of a 'patron card' and suspect it is a mistranslation. Patron, which comes from the Latin root meaning 'father', is not the same as customer. It is occasionally used for people who regularly attend theatre and opera, for example, because these (supposedly wealthy) people are supporting the arts by doing so.

Do you mean a card that can give you discounts and the suchlike? A loyalty card or a clubcard are what I would call it.

I'm a U.S. native speaker. I would call it a "loyalty card" or "discount card." I have not heard it called "patron card." If you're referring to a specific card for a specific store, it would be natural enough to use the store name--"I have my Supercuts card," meaning a card that gives me a discount at the Supercuts chain of hair stylers. If you pay a fee to become a "member," then it would be natural to called it a "membership card." I have a Costco membership card.

What type are yo thinking? There are many types of cards for like membership, discounts, store only cards, credit cards, debit cards, etc. Some features are combined. Examples: Sam's club and Costco have membership to use for getting gas for instance. Insurance cards have discounts for me on medications (meds). Victoria Secret, Dress Barn, Aeropstale have cards that you can use only in the store. Visa, MasterCard, discover are Credit cards. Bank cards are debit cards but can be used as a credit card sometimes.