If a credit card company issues a card in your name to someone else, then clearly they didn't do their verification properly.
It should be possible to have all your personal information floating around and still be certain that no one can use that information for anything. I'm pretty sure it works for Sweden where their SSN aren't secret, nor are addresses or phone numbers.
Sure that may mean that you need to show up in the bank, or have a secure government authentication method, but that's not beyond implementation.