The scheme was designed in 1935 as a way of identifying people enrolled in the then-new Social Security program (hence the name), and was gradually made a de facto national ID number over the intervening decades as the increasing scale of national population and economic activity prompted a desire for greater legibility to facilitate management and taxation.
Greater legibility, management, and taxation are values not shared by a very sizable section of the US citizenry. Without digressing into a pointless and stupid flamewar over Humean oughts around these points of fact, we can acknowledge that every attempt thus far to implement a national ID scheme has been, and no doubt any future such attempt will be, energetically and, thus far at least, very effectively countered by those who so strongly prefer that no national ID scheme exist. Unfortunately, the US federal bureaucracy being what it is, we will have a national ID scheme whether well-designed as such or otherwise, and the wide adoption of Social Security and (generally) 1:1 mapping between SSNs and citizens made it the obvious candidate for a de facto national ID. Businesses followed government in adopting it as such.
Now, instead of a proper, admitted national identification scheme which would at least have a chance of being marginally secure, we have a motley collection of the world's most hilariously shitty zero-factor authentication methods, making a wide variety of frauds so trivially simple to perform that we've had to spin them off into a new category of their own, "identity theft", which we regard as just one of those unfortunate things that happens sometimes, like cancer. Even after the Equifax breach, it'll
probably still be cheaper to continue looking at the problem this way than to roll back the eight decades of technical debt that have gone into creating it. So it goes.