I've written many posts about this topic, because many people have pets and some are preppers.

Obviously if you do care about your animal and you also want to be frugal, then you purchase their food when it's on sale and rotate it so they have plenty but save yourself money. Hey, that's real prepping not buying bugout bags...hmmm

Let's say you have a dog or cat. What do they eat in Nature? They're mammals that are predators and eat meat. In a collapse, you could easily harvest by trapping small rodents and supplement their diet from your pet food supplies. Trapping is the way our ancestors most often ate meat, not hunting which is way more demanding of calories used to produce work.

Many diseases are spread by pets from their fecal matter. For example, in a collapse, local water supplies could be compromised and people and pets would drink from contaminated streams or lakes. Pets could get amebic dyssentary or they could get toxoplasmosis gondii or they could get about six different varieties of worms. This means know how it's spread, making sure your pet drinks clean water, knowing how to find water and dig wells, not allowing their fecal matter to be composted and recycled in your garden, etc.

In a collapse, you might become their veterinarian. It's that or if they get sick, they'll potentially die. If you care about them, they maybe you download the veterinarian manuals about their specific needs and learn how you might best limit their exposure to common ailments and how you might treat them. You probably won't have medicines. How did our ancestors treat them? Learn by reading history.

It's that or putting them down when they get sick and potentially infectious and making other community animals sick. Do you really want to do that? Do you have the capability and know-how to do it so they don't suffer? If not, then it will be cruel when you have to do it, and make mistakes and they suffer worse because of your ineptness.

It's been hypothesized that many people will abandon their pets in a disaster based upon previous history with say Hurricanes or wartime. When that happens, animals go feral. Dogs will run in packs and share their diseases. Because a dog is a social creature that likes to be around humanity, but may have lost its normal fear and docility from taking care of itself, a feral dog has a high chance of attacking a weaker human in order to survive.

People worry a lot about wolf attacks, but there's little documentation about them. It's far more likely for a dog to attack, and especially when it has gone feral. Some species of dog will interbreed with coyotes or wolves too, and then become a very dangeous species to deal with.

These are the basic kinds of issues with dogs and cats in a collapse.