Hamburgers are a specific style of sandwiches whose name is derived from Hamburg, Germany.
Chicken sandwiches are not hamburgers.
A chicken sandwich is a sandwich when the chicken meat is between two slices of bread cut from a bigger loaf. It’s a chicken burger when it’s between two halves of a bun.
And as you said, hamburger derives from the German city of Hamburg, so Germans, not Americans, have the authority here.🤪