Only in recent years did parents have to get a newborn a SSN. I was born in 1960 and didn’t get a SSN until I started working when I was 16. I’m sure Obama was the same. His mother may have gotten a SSN even later in life. The SSN would be from the state where it was issued, not the state where they were born.
I remember when I was younger and that was a long time ago way before you and my mother had to get ss numbers for myself, my brother and sister. I believe we got them when we were 5 years old and that was way before 1960.