Why is a piece of chicken between to buns called a sandwich rather than a hamburger?

There is NO ham in hamburger.

A hamburger contains a red meat either beef or lamb. It would be called a chicken burger. A sandwich is made from sliced bread. You can also call it a chicken roll, or baguette depending on the type of bread used. If you use a pide or focaccia it would be a chicken pide or chicken foccacia.

The person selling the item may not have their sandwich game up to scratch and mislabeled it.

It's a sandwich because its chicken in between and not a burger.

The beef is called a hamburger because theres a burger in it.

The term hamburger originally derives from Hamburg, Germany's second largest city, from which many people emigrated to the United States. In High German, Burg means fortified settlement or fortified refuge and is a widespread component of place names. Hamburger, in the German language, is the demonym of Hamburg. Similar to frankfurter and wiener, names for other meat-based foods, being demonyms of the cities of Frankfurt and Vienna (Wien), respectively. The term "burger", a back-formation, is associated with many different types of sandwiches similar to a (ground meat) hamburger, using different meats, such as a buffalo burger, venison, kangaroo, turkey, elk, lamb, salmon burger or veggie burger.

the "chicken between bread" thing was not invented in Hamburg

At Red Robin, it IS considered a burger.

Idk, maybe cause it's not beef, good question