The army and air force discontinued issuance of military service numbers and began using SSN effective 1 July 1969. The navy and marine corps in 1972 and the coast guard in 1974. Your Social Security Number is now your student number, drivers license number, banking identification number, taxpayer number, veterans claim number, etc., etc., ad infinitum. It's not REQUIRED by law but the armed forces are not REQUIRED to accept you; banks are not REQUIRED to allow you to open an account; schools and colleges aren't REQUIRED to accept you as a student, etc.
There's light at the end of the tunnel, however. An increasing number of court decisions - prompted by a correspondingly increasing number of individuals re-discovering that they have rights after all - are gradually negating this unconstitutional policy. For example, an increasing number of states cannot use your SSN as your drivers license number without your expressed consent. No big deal? Anyone out there. Send me your SSN, nothing else. I can have your complete life, employment, credit, etc., histories in a matter of hours. Imagine what I can do with that info.