Bunny is well, a bunny, but if one read between the lines of some of the stories that feature her, her real name could be and mostly likely is Ethelinda Berniece Rabbitwright. Another subversion like the Linkara example being that 'Bunny' is a nickname.
Confirmed here that her real name is Ethelinda Berniece.
A Youtuber known as The Archfiend had a dog once. Some time after it passed away, he decided to get a new dog, which to this day is still known as 'New Dog'.