Hamburgers are named after the city of Hamburg, though no definite connection between the two has ever been drawn, from what I see.

In the mid-1900's, there was an effort to rename them beefburgers to make the ingredients more explicit (by this point, the shortened form 'burger' had already taken off, though it has no meaning outside of being a shortened form of hamburger), but those efforts failed.

Sandwiches made from meat patties other than beef are named with the format $meat + "-burger" because it is assumed that an unspecified burger is beef, so we need to specify.