Yep. That's why "loyalty cards" exist. Since they're not allowed to associate your purchases (or really any data) with your CC number to build a profile they give you a separate ID number that you key-in/scan when you buy things.
"Oh but you don't have to use your loyalty card."
Technically true but it's not "get a discount if you use your loyalty card" it's now "pay really inflated prices if you don't."
For what it's worth, I know people share loyalty cards across large groups to mess this up. Me, I just eat the cost. Developing a "I will not play your games" has been great. I know people who absolutely obsess over gamified consumption (e.g., airline miles) and I'm glad to have the brain space for things that matter.
I have to say I think loyalty cards are a distraction, why can’t they just track my via my card info/Apple Pay? I mean the thing is literally a kind of unique identifier...
"Oh but you don't have to use your loyalty card."
Technically true but it's not "get a discount if you use your loyalty card" it's now "pay really inflated prices if you don't."