In Denmark, a CPR number (short for Det Centrale Personregister, or Central Person Register) is a unique 10-digit personal identification and social security number assigned to every resident and citizen.
Equivalent to social security information in the US I guess.
It's unique, but it encodes your birthday and sex.
There's only 500 numbers it could be, assuming someone knows those other things about you.
In any case, there are alternative systems for authorisation.
I'm not super familiar with SSN in US of A, but I think the Danish one is not has secret. Don't get me wrong, the leak is not good, but there is a limit to what you can do with it.
Targeted and real looking spam mails come to mind. Hey <NAME> with <Address> and <CPR>, you have to log in <fake government website> to verify X Y Z.
Apparently some pay day loans or similar with just CPR + name is or was a thing. But lets hope that will change now. Bonkers as CPR should be be treaded as a secret.
For some unknown reason, the SSN in USA is assumed to be secret. You go to the bank, say SSN=12345 and they give you a million dollars and then send the collector the the guy/gal with that number, and call it identity thief instead of bad bank security or fraud.
Here in Argentina, the DNI is assumed to be public, it appears in a lot of public documents next to your name, and on election day there is a list of all the local voters with name and DNI at the door of the pooling site. To pay a sweater in two installments you may need to present the phisical DNI card and a water or electricity bill and they photocopy all of them.