So in the 17th century, the Bishop of Quebec approached his superiors in the Church and asked whether his flock would be permitted to eat beaver meat on Fridays during Lent, despite the fact that meat-eating was forbidden. Since the semi-aquatic rodent was a skilled swimmer, the Church declared that the beaver was a fish. Being a fish, beaver barbeques were permitted throughout Lent. Problem solved!
One of their origin stories is, that a monk created them so he can secretly eat meat on Fridays. Since the meat was cut very small and "hidden" in the dough between vegetables and other stuff, God wouldn't be able to see it. That's why it's also called "Herrgottsbscheißerle" in swabian which roughly translates to "small cheater on God".
The idea that fish is ok makes no sense. It makes sense from the fact that denying easy protein to a culture living in a desert on the ocean would NEVER fly. But not in any way for categorization.
Secondary Fun Fact: It was created by a franchise owner, against the wishes of CEO Ray Kroc, who had created a sandwich with grilled pineapple as a meat substitute for that purpose.