Having a foot in each of two islands that between them invented the sandwich and also have about 3,000 regional descriptors for a bread roll, none of which is “sandwich”, calling either a hot dog or a burger a sandwich is some kind of crazy witchcraft that needs stamping out!
I’ve never thought about a Burger being a sandwich. It’s closer than a hot dog that’s for sure, but it’s got its own category of a Burger. It’s a burger not a sandwich.