you just need to add their name while checking them out using the customer directory feature of Square and you will have a name tied to their transactions. Not all credit cards automatically transfer the name and by using customer directory you get many more features.
https://squareup.com/help/us/en/article/6217-get-started-with-your-customer-directory
https://squareup.com/help/us/en/article/5532-manage-your-customer-directory-from-the-square-app
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