The real problem here is the attempt to maintain permanent one-to-one mappings between ID numbers and humans. The legitimate purpose of a government ID is so you can e.g. go to the bank, open an account and then later establish to the bank that you're the same person who opened the account. If you want to get a new ID number and start over, you shouldn't have to steal someone else's in order to do that, you should just be able to go to the DMV or the social security administration and get a new ID under a new name that isn't already somebody else's.
The hypothesis that this would help criminals is pretty thin. They're already going to use an assumed name, which is why law enforcement uses photos/fingerprints/DNA to identify suspects rather than a government ID that people aren't actually required to carry regardless.
Getting an SSN for your child isn't compulsory, so the system also isn't expected to hold every person.
For the majority of people, it's 1-to-1. But it's not guaranteed that an SSN identifies a person (if it's been replaced) or that a person has an SSN (if their parents were lazy or are sov-cits)
"UNIQUENESS. The SSN is not a unique label. More than 4.2 million people, by the Social Security Administration's own estimates, have two or more SSNs. More serious, although much less prevalent, are the instances in which more than one person has been issued or uses the same SSN."
Forcing people to have a unique permanent barcode is primarily of use to authoritarians.