It is the second cat in Poe’s story that the narrator blames for the death of his wife.
Pluto, the first cat, is killed and then the narrator’s house burns down. When he sees a nearly identical cat at the tavern, he is compelled to take it home with him.
Slowly, the narrator begins to despise the second cat just as he grew to hate Pluto. As he and his wife descend the stairs into the cellar of their new, shabbier home, the second cat nearly trips the narrator. In a fit of rage, he swings an axe toward the cat, but his wife intercepts the blow with her arm. Even more enraged, the narrator instantly “buried the axe in her brain, and she fell dead upon the spot without a groan.”
The narrator inadvertently murders his wife, whom he professes love for at the beginning of the story, because of the second cat. One might speculate that without the second cat, the narrator never would have murdered his wife.
On the other hand, he says that the murder and concealment of her body “troubled [him] but little,” meaning he felt little remorse for killing his wife. Therefore, one could say that even without the second cat, the narrator eventually would’ve been driven to kill again.