The SSN is not a national ID number, it is being used as one, but that was never the purpose behind it.

Just the fact that it is a serial number without any checksum is enough to tell you that it is not a proper ID number system.

This quote from the below article shows how ridiculous the situation is:

currently citizens are identified by a patchwork of documents issued by both the Federal government of the United States as well as individual State governments of the United States and Local government in the United States governments.

https://en.wikipedia.org/wiki/Identity_documents_in_the_United_States

You never want to hear/read the word “patchwork” when describing a national ID system…