Right. I understand that. She's an American citizen according to the story. How does an American soldier earn a foreign badge on American soil?
Forces often offer specific badges to partner nations.
Some are field proficiency.
Some are weapons proficiency.
You sign up and get selected for a two or three day course, where you learn and demonstrate proficiency, and you’re awarded the foreign badge.
If there are Germans stationed in the US (there are), they can offer the course locally. It’s more inter-service/country PR than anything really meaningful.
Most of the badges cannot be worn with the uniform, except on special occcasions. If we earned a German badge and went to a party in Germany, we’d wear the badge on the uniform (permission granted).