I've had salmon burgers, turkey burgers, veggie burgers, venison burgers. So a burger doesn't necessarily need to contain ground beef (or even ground meat) to qualify. However, I am not certain what the limits are on what should earn the tag.
The existing description is only a couple sentences long:
Questions about the classic American sandwich, the hamburger and how to prepare them. Do not use this tag for "hamburger meat" aka ground beef or beef mince outside the context of the sandwich.
While questions about raw ground beef are excluded from the tag, nothing else is specifically left out. Existing questions in the tag are almost entirely about beef hamburgers, but I did see one about veggie burgers, so that didn't help me either. (For what it's worth I don't think hamburger is a great tag for the question I link to above, which is about a fried salmon filet with the same bread and condiments as a beef hamburger. Personally, I wouldn't even classify that as a salmon burger...)
Before editing the tag for a little more clarity, I figured it would be better to ask a meta question to establish how it really is expected to be used.
Alternate thing to consider from comments on an answer: renaming the tag to a somewhat more general burger, instead of relegating that to a synonym?