My question applies to the entire Western Classical Age, Roman dogs as well.
Note that I'm not a breed connoisseur; I'm curious about characteristics that would stand out to everyone from modern times.
As a rule when they are hungry the master should feed the hounds himself; for when they are not hungry they do not know to whom that is due; but when they want food and get it, they love the giver.
(Xenophon, "On Hunting" 3.7)
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...just as dogs bring on a vomit by eating grass and thereby purge themselves.
(Aristotle, "History of Animals," 7.20)
If there's anything comforting to be found in the study of history, it's that dogs are dogs. Or you might justifiably say, doggos are doggos. Because what tends to be of more interest to historians than "what was a dog?" is "how did humans understand dogs and fit them into changing cultural contexts." And there, the biggest constant is that dogs are important and people loved them. My favorite example of this is actually from the medieval Islamic world, where moralists and veterinary science writers scold dog owners for making clothes to keep their dogs warm at night out of silks and other expensive fabrics.
That's not to say there aren't differences between the ancient (and medieval) world and today in terms of human-canine relationships. Probably the biggest difference is evidenced by classification. Most people today, at least in the west, would classify dogs by breed first, and by that we first of all mean looks. We might pay some attention to their group insofar as it tells us whether we'll be taking our pet for a two-mile walk every day...or going for a two-mile run in the morning, at night, spending an hour on training, getting involved in at least one dog sport, and still realizing that the dog figured out you don't leave the house without putting on shoes so overnight hid all the shoes expecting this means you can't leave. But for the most part, "Toy Dogs" "Non-Sporting Dogs" "Terrier Dogs" are groups at Westminster.
On the other hand, early authors are much more concerned with function first, and breed second. Xenophon writes about dogs in a larger treatise On Hunting; Roman scholar Marcus Terentius Varro in On Agriculture. We also hear about war dogs in classical sources, including among the northern European "barbarians." (Even the dogs are barbarians, albeit the OG noble savage--"if you are not set on looks and deceptive graces," writes Grattius).
They are certainly keen on physical and temperment descriptions of what makes a good dog for hunting, and the concept of "breeds" is apparent even when there don't quite seem to be uniform standards for what the breeds are. One interesting thing to me is that while classical and medieval authors are obviously aware that dogs come in lots of colors, classical authors make it an important feature to consider when breeding. Xenophon says only multi-colored dogs are well-bred and suitable for hunting (a prescription that must have been somewhat well known, since later author disagree with him by name), and Varro wants farm dogs/herd guardians to be all white so as not to be mistaken for wolves.
It's much harder to see traces of dogs as primarily companion animals in written sources, although hunting and guard dogs certainly had that secondary function. The medieval European Livre de Chasse and Master of Game prescribe the dogs to sleep in a separate doghouse but advise hiring a kennelkeeper to sleep with them and keep them company--HIRE ME--but veterinary texts from classical India and the Islamic world have dogs sleeping in the house with people, right to concern about whether it is proper for dogs to sleep on the humans' bed.
Some scholars have posited this is due to a genuine change in relationship between human and dog--the idea of dog as primarily companion is Renaissance and later, with medieval puppers like Christine de Pizan's acting as a sort of status symbol/token of belonging at court. Another intriguing possibly is that we get a very slim perspective through male-authored literature. In medieval sources, at least, there are offhand references to women's lapdogs in the Near East being trained to perform tricks. And didactic writers say anchoresses can have a cat to kill mice and rats, but absolutely not a dog.
Although the ur-literary dog is unquestionably Odysseus' beloved Argos, the only one to recognize his master return home (upon which, having successfully completed his adopted mission to see Odysseus safe again, Argos can die in peace), dogs play a lot more roles in early literature than the Loyal Hunter--in fact, the sheer variety of dog "characters" in literary texts illustrates their ubiquity in human society. People interacted with dogs they liked and didn't constantly.
A hound that ignores the trail3 and races over the track of the hare on the run is ill-bred. Some, again, will pursue hotly at first, and then slack off from want of pluck; others will cut in ahead and then get astray; while others foolishly dash into roads and go astray, deaf to all recall. [9] Many abandon the pursuit and go back through their hatred of game, and many through their love of man. Others try to mislead by baying on the track, representing false lines as true ones. [10] Some, though free from this fault, leave their own work when they hear a shout from another quarter while they are running, and make for it recklessly. When pursuing some are dubious, others are full of assumptions but their notions are wrong...Now most of these faults are natural defects, but some by which hounds are spoilt are due to unintelligent training.
(Xenophon, "On Hunting" 3.3)
And this passage, I think, shows that classical people absolutely understood the heart of the human-dog relationship that you hear scientists discuss in the Nova and other documentaries: it is a relationship. Dogs comes in all sorts ("dog first, breed second, individual third"), and as Xenophon discusses previously breeding matters, but it is above all their interactions with humans--good and bad--that determine what kind of doggo they'll become.