It has nothing to do with a central bank and identity theft can be solved easily. The first thing that needs to happen is your credit profile is in a default frozen state. The second is that identity theft is treated properly as bank fraud. Poof, identity theft is nearly eradicated overnight.

In other countries, the government provides a reasonable authenticator for personal identity; the banking system is not so easily fooled.

It's not an essential property of central banking that banks will give you the money in an account, or courts will enforce a debt against someone, simply because you know some basic demographic information about the victim.