"Патрикеевна" is just a patronimic name from "Патрикей" (this name is almost not in use nowadays). No idea why a fox is commonly referred to as "Лиса Патрикеевна" (also, a bear is called "Михайло Потапыч" or "Михайло Иваныч").

Typically, a fox is considered as a "[over]smart ass", while a bear is a strong, friendly, and simple guy.