I interpret it as an adjective; there are already a variety of alternative burgers such as chicken or turkey burgers. They're typically meat, but I guess I've never assumed they had to be; just that most plant-based sandwiches are closer to other things. Interestingly, salmon in patty form is usually referred to as a salmon patty even after being cooked (at least around me).

I don't find it at all confusing. A veggie burger is a patty made of veggies, just like a turkey burger is a patty made of turkey. The only case where I think it might be confusing is if the patty is made out of something that could plausibly also be blended into the meat of a regular burger. If the box said "jalapeno burger" and inside was veggie burgers with jalapenos in them, I'd feel mislead.