Here in Uruguay (and almost everywhere else) we have a national ID number (Cédula de Identidad). It's not supposed to be secret (although it's not a great idea to divulge it freely).
The ID number in my country has photograph, fingerprints, signature, several security measures (difficult to fake). Not so easy to steal (of course there are forgeries, but identity theft becomes much harder)
I'm not from the U.S., but I've read a lot of criticism about the SSN, for example:
http://www.theverge.com/2012/9/26/3384416/social-security-nu...
"In 2009, researchers developed an algorithm that could guess an individual’s SSN with up to ten percent accuracy"
"SSNs have become available through data resellers, security breaches at various companies and government agencies, unsuspecting customer service representatives, and even public records, if you know where to look. SSNs can be bought in bulk for $1 each on private online forums, and a specific person’s SSN can reportedly be had for as little as $3.80."
I've also read about duplicates ("More than 20 million Americans have more than one SSN associated with their name."), the numbers running out (in the 2050s apparently), etc.
http://www.witn.com/home/headlines/111371029.html
The REAL ID act sounds like something closer to what we have:
http://en.wikipedia.org/wiki/REAL_ID_Act
Some other advantages (from an IT point of view): you can validate an ID number against a central database, and get a person's given names in a unique format. Our ID numbers have a check digit, so you can validate if the ID number has been correctly entered :)
Multiple identifiers per person is a feature, not a bug. People get new SSNs when they want to change their identity (eg, witness protection programs), or because someone stole their old one. That need doesn't go away just because you have a new numbering system.