IANAL, but I believe that something like this would require a constitutional amendment, unless it could be squeezed in under the umbrella of the ICC like most other federal overreach laws.
If one agrees that this is true, that's not an answer to the question of why not do it; it does affect the mechanics—either a Constitutional Amendment (to do it federally) or a fifty-state compact (do it as a coordinated state-ID system for all state purposes, which is arguably closer to what was described) would be available mechanisms.
“The Constitution doesn't currently allow the federal government to do it” is not an answer to “why not do this in a centrally coordinated way”.
Personally I think there is some constitutional justification for a national ID system. The commerce clause gets abused a lot IMHO but identity is at the core of just about any contract. You could argue that state identification and plain old name, address, signature are sufficient, but it's not like SSNs are ubiquitous for no reason. Even as dumb as it is to use a single number for someone that never changes, the benefits of a single static identifier that is the same across a person's lifetime and established by the government outweighs the horrific insecurity of it all. Fundamentally a government is the root authority of legal identity for all the citizens of that country. The government should provide a modern secure way of establishing identity for both public and private matters.