Bree developed a very high opinion of himself with only non-talking horses to compare himself to, further enhanced by his status as the prized steed of a Calormene lord and his training as a war charger. Furthermore, having spent less time in Narnia than most talking horses, he had certain incorrect notions of Narnian behavior and culture.
Bree believed highly of himself because he only interacted with non-talking horses, which was further supported by his status as the prized steed for a Calormene lord and his training in battle. Moreover, since he spent less time in Narnia than most talking horses, he had incorrect notions about Narnian behavior and culture.