Not entirely. Think stores are still free to use any signals available at their own premises, like camera footage, license plate numbers in their parking lot, Wifi Mac addresses, or even gait analysis if they feel like it :-/
This is why I use <local-area-code>-867-5309 for every "loyalty" chain. I'd rather get the discount but also mix my data with as many other people as possible to make it worthless.
as I mentioned, they can observe purchase patterns and derive that a given individual is making those purchases if its unique enough.
gently, my take on privacy is that, we will never be able to turn the clock back to 1970; we have to instead reconceptualize privacy in the age of pervasive sousviellance & surveillance.