More precisely, a U.S. consulate would have helped.  There is one on Okinawa, where most American military are stationed. (Not that many are near the Embassy in Tokyo.)  One website advises:

If you have access to a U.S. military post:

• You can go to the Post Adjutant or Personnel Office. 
• These offices can copy and certify your records so you don't have to send original documents through the mail. 

https://tricare.mil/LifeEvents/Baby/Resources/SocialSecurityCard

"They were told" might be true.  Or it might be a way to cover up, "We forgot to ask."