They’re associating it with your debit/credit card, unless you’re buying with cash only. Also, the “identity” isn’t so much the target as the “profile”. Don’t get me wrong, if they are able to personally identify you, the communication will be much more… personalized… (good english) - mailings, texts, emails and coupons for stuff either you’ve bought or adjacent to your stuff (with better margins for the store) addressing you by name, grouped with other purchase-history items. But back to the profile: building a profile of your likes, dislikes, brands and such is valuable data that gets more $ for corporate as the fidelity gets higher. And as it does, the messaging to you gets more targeted.