The second cat represents a number of things, including perhaps what might be considered an "inner demon" of the narrator. The cat resembles Pluto in that it is missing one eye but it also has a splotch of white over its breast suggesting a focus on the heart. This connects rather directly to the narrator's inability to feel or love after the ravages of his alcoholism have encouraged his murder of both his beloved Pluto and then his wife.
It also connects to the habit of the cat to climb up upon the narrator's chest so that he feels its weight physically upon his heart. This increases his inner torment and the heavy weight of the guilt he feels.
His loathing of the cat is mirrored by its greater affection for him. This too suggests a strengthening of the guilt, the inner demon inside of him. His dread of the "beast" mirrors the crushing weight of the guilt he feels.