The free rider problem, SSNs and PANs
The free rider problem has been studied extensively by economists. We get the free rider problem when we have a good or service that's free to others once it's been paid for once. If this happens, then the people who get the good or service for free have no incentive to pay for it.
This why, for example, that some things are paid for out of taxes instead of being privately funded. National defense is the most commonly-used example of this. Once you have a system of national defense in place, it protects you whether or not you were one of those that paid for it. Because of this, it would be hard to create a workable system of privatizing national defense because of the strong incentive for people to have someone else pay for it while still getting the benefits from it.
It certainly looks like Social Security numbers are another example of this. The government has created a unique identifier for people. It's hard and expensive to create a system that creates and manages such unique identifiers, so there's a strong incentive for others to use an existing unique identifier instead of creating their own.
Credit card numbers are yet another example of this. Many of the parties who process a credit card transaction need a unique identifier, and the credit card number is a handy one that happens to be readily available. And just like there's absolutely no reason why you need to use an SSN to identify people, there's absolutely no reason why you need to use a credit card number in this particular case. But because it's hard and expensive to create an alternative, there's no incentive to create an alternative.
Economists have proposed a number of ways of dealing with the free rider problem. Maybe some of them could be applied to Social Security numbers and credit card numbers. If we could eliminate the use of those numbers in inappropriate situations, we'd also reduce their exposure to cyber-criminals, and that's probably a good thing. The big question is which of the solutions that economists have proposed would actually be workable in the real world. I'm not sure that anyone has thought about that particular question much.