You have to realize that the entire concept of "identity theft" comes from having centralized identity to begin with, otherwise there is nothing to "steal".
Suppose you want to take out a mortgage on a house. If you take it out in someone else's name, this is a problem. But suppose that didn't even enter into it. Instead you prove title to the house, i.e. you authenticate to the city title office as owner of that property using the authentication method you established when you bought it, and that proves to the bank that you own the property. You, having authenticated to the city, approve the bank to take a lien out on the house. They accept the lien as collateral for the mortgage loan, and you get a mortgage loan. Your name doesn't enter into it at all, so nobody could use your name to take out a loan. If you don't pay the loan, they don't care one bit what your name is, they just foreclose on your house.
That's one identity, but the owner of the house would have other identities. The fact that you know that the owner of the house approved the lien would not automatically tell you that, for example, the person living in the house approved the lien. Or that a certain employee of a certain company approved it. These would all be separate identities, even if they all refer to the same person.
Even in countries with unique, centralized identities, you don't go around handing your government ID to everyone you meet. You use it for official legal business only. In other contexts you still have less formal identities which remain separate from your official identity.
Suppose you want to take out a mortgage on a house. If you take it out in someone else's name, this is a problem. But suppose that didn't even enter into it. Instead you prove title to the house, i.e. you authenticate to the city title office as owner of that property using the authentication method you established when you bought it, and that proves to the bank that you own the property. You, having authenticated to the city, approve the bank to take a lien out on the house. They accept the lien as collateral for the mortgage loan, and you get a mortgage loan. Your name doesn't enter into it at all, so nobody could use your name to take out a loan. If you don't pay the loan, they don't care one bit what your name is, they just foreclose on your house.