There is probably not a connection. Death has two goldfish named Slim and Wandsworth. Dream has ravens, such as Matthew. Destruction keeps dogs, as we will see. Desire does not seem to have pets.
Despair has rats.
Note Despair's use of pain to distract her from unpleasant topics. Rats, however, are generally credited with first bringing bubonic plague to Europe. There were three forms of plague in the epidemics. Peasants and clergy were the most affected. The London Plague of 1665 claimed more than 70,000 lives out of a population of about 500,000.