Death's Dog is Death's dog. He collects canines when they pass away. He shares the same voice as Death, but unlike Brian, stands on four legs.
Death's Dog has only appeared in one episode so far, Mr. Saturday Knight, where he arrives with Death thinking that he needs to take Brian, who temporarily choked on a dinner roll.
Death's Dog: Hey, somebody choke on a roll in here?
Brian: Oh no, I spit it out.
If the theory is that he and Death collect the souls of their respective species, then this might contradict what Chris says to Brian in FG401 "North by North Quahog." The dialogue went as such:
Chris (to Brian): I don't have to listen to you! You're just a dog, you don't have a soul!
However, as Chris is an unreliable narrator, he probably does not state a rule of the Family Guy universe.