Find an animal, find the disease (some are vector born, some are directly from the animal). The best way is to medline the animal in question and limit to human disease.
Sleeping with their pet, which the majority of pet owners do, as an important way to spread numerous animal associated infections (CDC).
Pets, but animals of all kinds, can be a source of multidrug resistant human pathogens: S. aureus, E. coli etc. Every time I see a dog in the hospital I get the heebie-jeebies. Remember. To be blunt, dogs and cats lick their butt, then lick everywhere else.
The US imports 250 million weird-ass animals each year, each with their own infection risk (PubMed). it is my understanding the illegal drug trade is bigger than the drug trade. Imagine what would happen if people could get high by smoking Koala's?
If for some odd reason you are going to fraternize with animals, the CDC has advice on how to avoid infections. Go CDC.
The why behind zoonoses is complex and relates to more than pet ownership: there are issues with farming and environmental change (Ref).
A Partial List of Animals and Their Infections
Cat: Bartonella (cat-scratch disease), Campylobacter, Corynebacterium ulcerans, Cryptosporidiosis, Enterohemorrhagic Escherichia coli, Giardiasis, Pasteurella, Plague, Q fever, Streptobacillus moniliformis, Spirillum minus, Salmonella, Toxocara, Toxoplasma, Tularemia.
Cattle: Babesia, Bacillus anthrasis, Brucella, Campylobacter, Cryptosporidia, Enterohemorrhagic Escherichia coli O157-H7, Erysipeloid, Leptospira, Listeria, Lyme, Pasteurella, Plague, Q fever, Rabies, Salmonella, Toxoplasma, Tularemia, West Nile encephalitis, Yersinia.
And, while not infectious, 25 people are killed by cattle each year (CDC).
Dog: Brucella, Bacillis anthracis, B. zoohelcum, Campylobacter, Capnocytophaga canimorsis, Cryptosporidia, Corynebacterium ulcerans, Enterohemorrhagic Escherichia coli O157-H7, Erysipeloid, Giardiasis, Leptospira, Lyme, Pasteurella, Rabies, Rocky Mountain spotted fever, Salmonella, S. canis, Toxocara, Tularemia, West Nile encephalitis, Yersinia.
Many hospitals have 'pet therapy.' Everything in the hospital is a therapy. I think it is stupid to have dogs (except seeing eye dogs) in the hospital as 1) they bypass hand washing and 2) when cultured they are chock full of pathogens including Clostridium difficile, Extended-spectrum beta-lactamase Escherichia coli, extended-spectrum cephalosporinase Escherichia coli, Salmonella, Pasteurella multocida or Pasteurella canis, Malassezia pachydermatis, Giardia, Toxocara canis and Ancylostoma caninum.
Household dogs and cats can serve as a reservoir for pathogenic E. coli and subsequent cystitis (PubMed) and MRSA. How do humans get it. Hate to tell you this, but dogs and cats lick their ass and you pet them or let them lick you.
Goats and Sheep: Brucella, Bacillis anthrasis, Campylobacter, Cryptosporidia, Enterohemorrhagic Escherichia coli O157-H7, Erysipeloid, Giardiasis, Leptospira, Listeria, Psittacosis, Q fever, Rocky Mountain spotted fever, Salmonella, Toxoplasma, Tularemia. Viral encephalitis, West Nile encephalitis, Yersinia.
Guinea Pigs: Streptococcus equi subsp. zooepidemicus.
Monkey: Herpes B. HTLV-1 from African ape bites.
Raccoons: Baylisascaris procyonis.
Rodents (rats, mice, ground burrowing): Streptobacillus moniliformis, Spirillum minus, Campylobacter, Cryptosporidia, Leptospira, Lyme, Lymphocytic choriomeningitis (hamsters, mice), Pasteurella, Salmonella, Tularemia, Yersinia pestis.
Seals: Sealpox (PubMed). Can infected those who rehabilitate seals, and who knew seals could be drunks.
Swine/Pigs: Flavobacterium IIb, Bacillus anthrasis, Bartonella (US SW) (PubMed). Brucella (If you are hunting feral swine in the south, and kill one, the preparation of the carcass can lead to infection (MMWR)), Campylobacter, Corynebacterium ulcerans, Cryptosporidia, Enterohemorrhagic Escherichia coli O157-H7, Erysipeloid, Giardiasis, Hepatitis E, Influenza A, Leptospira, Listeria, MRSA, Rabies, Salmonella, Streptococcus suis, Tularemia, Yersinia (ground burrowing rodents).
Whales: Erysipelothrix rhusiopathiae.