You're not even REQUIRED to have a Social Security Number now. For anything (you'd be amazed at the number of lawyers who aren't aware of that). The Social Security Act was passed in 1935 and implementation began two years later. The program was barely four years old when the U. S. entered World War II. Service numbers were assigned for identification purposes (the original Social Security Act specifically prohibits the use of the SSN for general identification purposes but that's just another law that's enforced or ignored "at the convenience of the government").
Just yesterday I responded to a message from a woman (?) in Belgium pointing out how so many today observe policy or practice under the mistaken belief that it's law. There's a big difference.