They have done this even before everything was Appified, at least in the US. My local grocery store used to give out customized bar code key chains. They scanned the barcode when you started checking out to tie it to your account, and apply the discounts.
Now, my store's app lets you build a list online, and when you tell it what store you are going to it sorts the list by aisle. I now use that exclusively. It saves me so much time. I find that useful, and worth the exchange of my tracked shopping habits.
However, I refuse to install their app on my phone, and interact with the list through their website. I feel there is just so much more tracking they can do with an app on your phone vs. a website login.