Dogs and cats are carnivores. We don’t eat carnivores in general, regardless of domestication.

The problem with eating carnivores (and to some extent, omnivores) lies in the diseases they can carry (pork for instance has to be inspected for trichinosis), and accumulation of environmental toxins in their meat due to their position at the top of the food chain. Also, they tend to be more expensive to raise due to their food requirements.

The further up in the food chain an animal resides, the more scrutiny its meat needs in order to be safe to eat. For carnivores, that's usually not worth the effort, as there isn't all that much meat in a domestic cat or dog.

The only odd one out are horses, who we don’t eat in spite of them not being carnivores.

That's really very much a cultural thing. In many cultures, horse meat used to be poor man's meat, because you need to get rid of old horses unfit for work in one way or another and not eating the meat would be a waste, but meat of old animals usually isn't necessarily of the best quality, so it was sold for relatively cheap. Over time, with declining reliance on horses as work animals, the availability of horse meat has declined and it turned into somewhat of a rare delicacy. Especially since the growing leisure horse industry doesn't contribute a whole lot of horse meat fit for human consumption due to the medications used on leisure horses accumulating in the meat and not being exactly healthy for people.

If you remember the European horse meat scandal (a discount supermarket chain had been selling "beef" lasagna that contained horse meat instead of beef, the problem there wasn't that the product was in any way unhealthy or dangerous, it was quality horse meat mislabeled as beef. It really was just a matter of deliberate mislabeling.