The problem is that SSNs weren't designed as a unique identifier for US identity, with no consideration given for protection, authentication, etc. The earliest SS cards came with 'NOT FOR IDENTIFICATION' on the front, was used for nothing except SS, and was in general designed as nothing more than a record-keeping number.

Then people figured out that since the government went through the trouble of assigning every* [1] american* [2] a unique* [3] number* [4], they might as well use that as their own ID too for inter-operability between different financial firms, and that's how we ended up in this shitty situation.

Every time somebody proposes 'hey we should come up with a better national ID number' you get people yelling about mumble mumble big government mumble mumble privacy mumble mumble feds tracking you around, and the proposal dies in committee.

[1] originally only issued to those who worked, and was not exempted from the SS system, notably children. Of course these days children have SS numbers because they're a de-facto ID number, and it's required to claim dependents, and with varying degrees of resistance from weird groups like the Amish. Additionally, non-eligible people are issued tax IDs instead, which have the same format as SSNs because it was easier that way

[2] big(?) political fights over who exactly must get numbers, of course, from religious groups, illegal immigrants, and many more

[3] except of course it's not unique, there were errors where people got duplicate numbers, and people can get reassigned numbers, thereby having more than one

[4] did you know they used to assign SSNs sequentially? The guy who was next in line got your number + 1