I'm Lucy, and I'd be happy to answer your questions today.
There are different types of these registers, but they all tend to have an option where you can choose to have your receipt emailed. If you've ever agreed to have a receipt emailed while using this type of device, it stores just that piece of information and ties it to your credit card number. There's actually a fast food place near me that emails my husband every time I buy something, because his email is tied to our credit card in their system.
Email address should be the only piece of information they have. One of the most common systems I see at cash registers is Square (which is still done through an iPad or iPhone), and you see them commonly at privately owned fast food places (including franchises of large chains), taxi cabs, farmer's markets, and other small businesses that traditionally only accepted cash. The merchant doesn't see your email address. It's done by the system the processes the credit card receipts. Here is more information:
Look to see if the receipt has a link at the bottom that says something like "Manage preferences for digital receipts." You should be able to tell the system not to send you any more electronic receipts unless you type in your email address and ask for one.
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