(This is a nice /r/AskAnthropology question, by the way.)

It depends on the animal, really; and it's harder to make quick guesses than you think. Sometimes, the process (not to be confused with how a culture actually initiates or allows the process) of domestication is relatively simple -- typically the case with smaller livestock such as birds. Turkeys, for example, seem to have domesticated two and a...half times: once in Mesoamerica about ~2,300 years ago^1 , another in the American Southwest around 3 centuries later^2 , and tentative evidence of a third domestication event in Tennessee around 1200-1400 A.D^3 . Of those three only the Mesoamerican lineage survives, so you could say that all domestic turkeys today hail from a single common ancestor or ancestral group.

On the other end of the spectrum is cattle, whose ancestral species was rather terrifying and not to be trifled with. People, however, seem to have done it anyway at least twice, possibly three times, with separate lineages we can identify. The line Western culture is familiar with are the taurine cattle, which were domesticated in the Near East roughly 10,000 years ago -- through DNA testing, all taurine cattle have been found to descend from a population of 80 cows at about that time^4 . The others are indicine cattle (or zebu) which were domesticated in India from an Indian aurochs subspecies, and I think the jury's still out on if African cattle are a variant of taurine cattle or their own independent domestication.

A number of animals, however, seem to have only been domesticated once. Dogs, as you ask, probably the most familiar domesticate in our lives (in one piece, alive), seem to have ultimately hailed from a single event in Asia^5 . Likewise with cats in the Near East^6 . Rabbits were domesticated by medieval European monks^7 and might have had a single population group if they brought in semi-domestic penned Roman rabbits instead of just bringing in rabbits from outside the monastery. Yaks also had a single domestication event in Asia ~5,000 years ago^8 .

How homogenous were domestic animals in the ancient world? Did breeds exist in any sense of the word?

Landraces certainly existed throughout the world. Some notable examples are the various sighthound-like breeds in ancient Egypt. Already you can see a large degree of variation; the development of 'breeds' at this point is still more or less in the hold of natural selection (though aided by the discretion of humans) as their use begins to dictate their shape. The basics of selective breeding only show up in historical records around 2,000 years ago^9 , and the intensive application of in-depth knowledge about it to deliberately create new breeds only began to really pick up at around the 18th century; this trend began with [Robert Bakewell](https://en.wikipedia.org/wiki/Robert_Bakewell_(agriculturalist%29) who essentially spearheaded the art, creating numerous breeds, and others would quickly follow. Charles Darwin would later cite his work as an example of selection.

References:

^1. Thornton, Erin Kennedy, et al. "Earliest Mexican turkeys (Meleagris gallopavo) in the Maya region: implications for pre-Hispanic animal trade and the timing of turkey domestication." PLoS One 7.8 (2012): e42630.

^2. Lipe, William D., et al. "Cultural and genetic contexts for early turkey domestication in the Northern Southwest." American Antiquity 81.1 (2016): 97-113.

^3. Peres, Tanya M., and Kelly L. Ledford. "Archaeological correlates of population management of the eastern wild turkey (Meleagris gallopavo silvestris) with a case study from the American South." Journal of Archaeological Science: Reports 10 (2016): 547-556.

^4. Bollongino, Ruth, et al. "Modern taurine cattle descended from small number of Near-Eastern founders." Molecular biology and evolution 29.9 (2012): 2101-2104.

^5. Botigué, Laura R., et al. "Ancient European dog genomes reveal continuity since the early Neolithic." Nature Communications 8 (2017).

^6. Driscoll, Carlos A., et al. "The Near Eastern origin of cat domestication." Science 317.5837 (2007): 519-523.

^7. Whitman, Bob D. Domestic rabbits & their histories: breeds of the world. Leawood, Kan., Leathers Pub., 2004.

^8 Han, Jianlin. "Yak: Domestication." Encyclopedia of Global Archaeology. Springer New York, 2014. 7939-7941.

^9 Lush, Jay L. Animal breeding plans. A Book Of The Iowa State College Press, 1947.

The major agricultural animals (pigs, chicken, cows, and sheep) were probably all domesticated multiple times, with pigs having potentially up to a dozen separate domestications (though likely less). The picture is less clear with other animals like horses and dogs. With horses, we're reasonably certain where they were domesticated (West Central Asia, probably Khazakstan but possibly as far west as Ukraine), but the jury is still out on whether there was one domestication event with substantial wild interbreeding or multiple independent events. With dogs, the situation is less clear and I'm not familiar enough to answer further.

Breeds would also have been known in the ancient world, although the firm boundaries we see in pet breeds today would not have been delineated in most places. Instead, proto-breeds were kept apart by property rights, distance, and selection. In interstitial areas like Spain, hybrids did occur, often forming distinctive new breeds that became recognized independently.