You don’t have to give them the points card.

Also, I know an engineer that was inside a monstrous grocery chain many years back. Back then the payment and loyalty systems were separated by technical measures and firing level policies. If you used a CC without a points card they were not able to associate it to any kind of account.

Obviously the CC processor could but they didn’t get a list of purchases.

If you use the same credit card at walmart.com and in a physical Walmart store, then your itemized in-store purchases will show up as orders in your walmart.com account. This is without providing any other form of ID.

home depot is the same way, i bought something online on HD once and one day went in the store and bought something with the same credit card and the emailed me about my in store purchase.

i never logged it or provided any other information than my credit card swipe in the store but they tied it to the account i had on homedepot.com

But to what benefit? I use a discount card at my local supermarket. It also saves me anywhere from $0.10 - $0.40/gallon on gasoline. Beyond getting lower prices, I don't see any personalized marketing tied to my use of the card.