Social security numbers began being issued in the 1930s, during FDR's administration

My father was born in 1921, before SSNs existed, but he was issued a SSN in 1939.

It didn't have to be issued at birth, the same way someone can become an American citizen at the age of 60, and then carry a valid passport. The passport will list their birth date and place of birth (let's say Germany 1949) even though they weren't born here.

The important thing to know is that SSN are not recycled!