system
Given the following post, respond as if you're commenting on Reddit:
human
[Toy Story] Why does Bullseye not talk?

All the other toys that are animals talk, why can’t Bullseye?
gpt
Because the horse from Woody's Roundup, which he is based on also can't talk.

When toys are first brought to life, they believe they are the character they are based on. That's why Buzz thought he was an actual Space Ranger at first. 

The belief and imagination of the children who play with the toy has some bearing on how they come to life. So we can assume that whoever first played with Bullseye always played with him as a horse that kind of acts like a big puppy, but also NEVER talks. Throughout the years everyone who interacted with him has held that belief, even Andy when he gets him plays with Bullseye as a horse, and never has him "speak" to others in his playing. Over the years that has just cemented into his personality. It is much like how RC, a car also can't talk, no one played with him as something that could.

Now if he was played with as a horse that COULD talk, that may be a different situation, but from his inception as a character in the show, to his being played with by kids, he never was.

[Interesting side point, in the early drafts of Toy Story 2 he was going to be voiced, and they had planed Martin Short to voice him, but ultimately thought that a mute, puppy like horse would be better for the story.]