A large part of their essay is just concerned with arguing against treating pets as property. I think it's pretty clear that the ordinary social meaning of having a pet already rules this out. One may carve up one's property for fun; if someone were to carve up their pet, we would (rightly) want them to be locked up for animal cruelty. If the legal system failed to do this, they would certainly be shunned by the rest of society, who would be deeply horrified by their actions.
It's an interesting question whether non-rational beings can have a right to life in addition to a right against cruel treatment. If so, the implications would be quite radical, even aside from the complete abolition of the meat industry. Society would presumably be obliged to support animal shelters to an extent that removes the current need to kill many perfectly healthy animals due to overcrowding. I think that's a plausible enough position, though there are counterarguments to consider.
Where the authors go off the rails is when they suggest that "domestication itself raises serious moral issues irrespective of how the non-humans involved are treated" -- such that pet ownership would still be wrong even if animal rights against cruel treatment and convenience-killing were secured. Why do they think this? What further rights are being violated, merely by caring for your pet? Here is what F&C write:
Domesticated animals are completely dependent on humans, who control every aspect of their lives. Unlike human children, who will one day become autonomous, non-humans never will. That is the entire point of domestication – we want domesticated animals to depend on us. [...] We might make them happy in one sense, but the relationship can never be ‘natural’ or ‘normal’. They do not belong in our world, irrespective of how well we treat them. This is more or less true of all domesticated non-humans. They are perpetually dependent on us. We control their lives forever. They truly are ‘animal slaves’. Some of us might be benevolent masters, but we really can’t be anything more than that.
"Slavery is bad, X is like slavery, therefore X is bad" is superficial reasoning. Much depends on whether X shares the relevant features or preconditions that explain why slavery is so bad.
I take the basic problem with (human) slavery to be that it is so drastically contrary to the interests of the enslaved. Not only were slaves historically mistreated in all sorts of ways, but even an imaginary "happy slave" seems in a tragic position insofar as their capacity for rational autonomy -- and hence for a fully flourishing human life -- is being stunted rather than nourished. Rationally autonomous beings have an interest in developing and preserving their autonomy, and when this interest is violated their life is (in this respect) worse as a result.
This crucial feature is obviously lacking in non-rational animals. So long as we do not mistreat them (whether by outright cruelty or mere neglect, e.g. failure to provide a sufficiently stimulating environment) domestic animals' chances at a fully flourishing life are not impaired by the mere fact of our control over them. They have no interest in being free of our control, because they have no capacity for rational autonomy that would be served by such "freedom". Life in the wild is often nasty, brutish and short. We can provide much better lives for our companion animals.
Perhaps the simplest way to refute F&C's argument is to note that moral rights must track interests. It makes no sense to posit a right that serves no possible interest. F&C acknowledge that domestic animals have no interest that would be served by a right against dependency or human control / guardianship. (They do not advocate abandoning domestic species to the wild, but rather taking care of those that currently exist whilst preventing any others from coming into existence.) Domestic pets are not better served by non-existence than they are by living happy lives under human guardianship. So it is absurd to posit a right whose only purpose would be to push us towards this worse outcome. That would be the most pointless, counterproductive right ever.
We do better to realize that dependency is not always and everywhere intrinsically bad. Dependents are vulnerable to abuse, and extended dependency may be an obstacle to important kinds of development, at least for beings -- like humans -- with the capacity for rational autonomy. But it is hard to see why dependency per se should be considered a bad thing for a dog. To think this would seem an unfortunate kind of anthropomorphism.