Not really early Germanic though, because every other Germanic language kept the meaning of "animal". It's only modern English (since the 1500s) that narrowed it all the way to one specific species (or family of Cervidae).

My guess would be that the language gained the word "animal" from French and "deer" was pushed from its niche and forced to specialize?

The modern English word "bear" originally came from a proto-Germanic word meaning one of "brown one" or possibly "wild animal". There was an actual name for bears, but speaking it was taboo in case it caused a bear to appear, so the euphemism eventually replaced the real name.

When I learned this originally, I was taught that the true name was lost to time, but Wikipedia just says it was "arkto" so whatever.