Sure. Problems arise when Social Security numbers are stolen because it's used as a de facto national ID number. In combination with your birth date and mailing address, possessing that is sufficient to "prove" identity to lots of official and quasi-official agencies.

One alternative would be to use an independent account number for each of those separate systems. That means more account numbers to keep track of, but in mitigating identity fraud that's a feature.

Biometrics help in certain applications, but trying to combine them with an all-authoritative national identity is a recipe for disaster.