I'm just speaking from my own experience.  Most Amish would "register" their births.  In fact, after Lovina and family moved to Michigan she needed birth certificates for her children to enroll them in the local school.  She wasn't sure how to get them, so I assisted her by rooting around the court house in Indiana in the county where her kids were born and was able to get those birth certs for her.

And, yes, the Amish do have social security #s.  In past instances in the US when there were drafts, Amish men were not exempt from the draft, but  - as conscientious objectors - they were permitted to serve in domestic wartime capacities such as hospitals, factories, etc.  This didn't sit well with many Amish since it was still aiding a wartime effort, but it was better than being forced to fight......