I don’t think of a horse as a luxury/hobby item, like a yacht or a kayak or a set of skis, because you can’t just dry-dock it or put it in the storage locker for the off season.
I think the contrast you are seeking is between pets and livestock, beween animals valued for their unique individuality as a companion, and animals seen in a purely utilitarian way (broiler chickens being maybe the extreme example). Different animals in the same species may exist at different points on the pet/livestock continuum.
For instance, our city has recently approved backyard chickens, but you are not allowed to slaughter them. If they fall ill, you must have them euthed by a vet. BTW, I am waiting for the next avian flu panic. Last time (before the backyard chickens existed) the ag dept went around slaughtering and burning carcasses at all the big local egg producers, plus rural backyard pet emus and fancy poultry. Anyhow, a sure set-up for conflict over the pet/livestock divide.
One difference is that, these days, cats and dogs live with us, so they do become bonded to our home lifestyles in ways that horses do not. Especially a horse in a full board barn. It could change owners several times and not experience much change in its life and routine, compared to a horse with one owner that moved between facilities.
That said, attitudes to cats and dogs have changed a lot since I was a kid. One of my friends (my age) often points out that 40 or 50 years ago, people would leave their barn cats and barn dogs behind when they moved, because the animals were “used to the place” and they thought they couldn’t be moved happily.
One of the reasons people don’t sell adult cats and dogs is that, except in the case of high-end breeding stock or maybe some performance dogs, there is no re-sale value for adult animals. Everyone wants a puppy. But people do give away cats and dogs all the time, either to friends, on CL, or to the shelters, if their circumstances change.
And the basic maintenance costs of keeping a cat or dog are pretty low, compared to keeping a horse. Yes, I know if you feed organic raw meat and get all the shots, teeth cleaning, grooming, and doggie daycare, house pets can be expensive. But they can also lead long, happy lives on bulk kibble, if the rest of the environment is good. So while people often have to rehome cats and dogs if they are moving to an apartment where they can’t keep them, they rarely rehome them purely for financial reasons.
Also dogs have very short lives, compared to horses. Who would buy a ten year old dog?
Horses fall on the spectrum between livestock and pets. Cows and pigs are pretty much interchangeable, and tend to be fattened up for slaughter. So we don’t expect to have long relationships with them.
Interestingly, even the horse breeders who are on the “livestock” end of the spectrum, even the ones that ship their “culls” or their injured horses to slaughter, don’t eat them, themselves. No one shoots a racehorse with a broken leg and puts it in their own freezer. So there are certain taboos about horses even on the livestock end of the spectrum.
As far as the pet end, well, the one-horse owner is going to have more emotionally invested in that one horse, than someone with 100 horses on their ranch has in any one of most of their horses (even there, the rancher will have his favorite riding horses that he has a special relationship with). But there are limits with the “pet” spectrum also, since you can’t really bring horses into the house (I did see a TV item once on a seeing-eye mini).
On the other hand, I expect that staying in full board would lessen the “pet” feeling even for many one-horse owners, since IME a big part of the relationship with the horse evolves out of doing that daily work, the handling and the feeding.
