Farm-Raised Venison and Bison Before the Common Era

Yes. I think it would be both, geography and characteristics and a whole lot of other things.
If I remember correctly, most domestic animals [other than the chickens] can be traced back to west-central Asia. That doesn't strike me as Bison or deer territory. You work with what you've got.
Why go after the Bison if the Aurochs was closer and more compliant?
Why tame the deer if sheep and goats are easier to herd, give you more milk, meat, wool and are on your doorstep?.
The Sami people have [semi] domesticated the Reindeer [Caribou] over the last however-long to make them more docile and compliant by killing off the more aggressive males. So why fully domesticate an animal if you can get what you need from something that looks after itself most of the time?

 

You also need to be able to fence them in. Building a fence that a bison can't push over should be quite challenging. And I don't know how deer would do confined to a pasture. Some zoos keep deer, but those have rather large areas to themselves that generally aren't just an open field covered in grass or dirt.

Another important factor is how much work you have with each animal to make it compliant. You can train one or two zebras, but it just isn't economical to train thousands to provide an army with steeds. Horses and camels are much less trouble in that regard.
With enough time and labor, there shouldn't be any evolutionary barrier to breed deer with pleasant demeanors that don't try to break out. But what farmer would invest all the needed resources if the process takes so long even his children won't ever see the results?

For fantasy, I think it much more practical to simply have another made up branch of a species that is naturally agreeable to domestication. We already have horses that seem almost identical to zebras but with enough small differences to make them candidates for domestication. No reason to not have a kind of deer existing in the world that is ready for domestication.

 

Personality would seem to be the leading contender, regardless of location. Eurasian peoples had a milder alternative to bison and focused their domestication efforts on it. Plains natives didn't have an alternative and not one tribe managed to domesticate the American Bison; since they domesticated dogs and eagerly adopted horses after their introduction, there obviously wasn't any sort of deficiency on the human end. Some Europeans tried and noted the creatures' foul temper as a reason why they failed.

There are American Bison ranches today, but these animals are heavily interbred with domestic cattle and remain pretty aggressive.

 

I can partially answer the "Why Some and Not Others" question: Size.

Bovines were domesticated because, ancient breeds were SMALL. The further from India westward you traveled, the smaller they got. Like, slightly larger than a large golden retriever or maybe the size of a Saint Bernard. Google ancient or heirloom cattle breeds. It's astonishing to see what a large cow *used to be*.

Human populations at some point in pre-history in the Indian subcontinent, Eurasian Steppes into Russia, Arab peninsula into Northern Africa, and Europeans inherited a bacteriological or genetic mutation that allowed them to keep digesting milk past infancy. Anthropologists have conjectured that this may be what gave early man an edge vs. other hominids, or certainly is our most recent adaptation as a species. Most younger and mature mammals are naturally lactose intolerant once they stop nursing from their mothers. That's why dairy and dairy foods were not really ever a 'thing' in some parts of Asia and the Americas until Europeans and Silk Road travelers were on the scene. They had the option to domesticate all kinds of mammals for dairy, and for some reason did not. I'd wager you would not be interested in developing dairy stocks if said dairy made you violently ill.

Temperment plays some role in the animal, but size matters more. We've tolerated goats and sheep for 1,000s of years and they can be ornery ill-tempered little bastards. But, the benefits of fiber, meat, and dairy encouraged domestication of an animal we could outwit and outmaneuver. Every generation gets a little bit tamer, especially if you selectively breed a good temperment into your wild captive stock. With huge animals, or animals with pointy-stabby things that regrow themselves and can leap like gravity isn't a thing, you'd probably go for the smaller but ornery creature that can't kill you as quite as easily.

The ranchers who raise bison around me (and the ostritch/emu ranchers) have to usually roll out in caged 4x4s or heavy duty 2 ton trucks to interact with part of their herds/flocks.

Yet, some bison are so tame in spite of their size, they're residents in childrens petting zoos.

I would also wager that pre-historic peoples tried to domesticate all sorts of creatures. At some point, it's also simple production mathematics: so many pounds of 'feed' is required to produce X amount of product (finish slaughter weight or dairy output). Some creatures just don't have a decent conversion rate of feed:roduct to make intensive domestication worth while. Goats are still the winner, by the way. It takes very little unspecialized feed, and little space, to produce high quantities of milk.

Why Caribou? Well, because they're still kind of on the small side and got used to having human handlers quickly. They're too busy roaming huge distances for food under snow pack that they probably were aware that people were not wolves or bears, so they could be tolerated. I wouldn't categorize Caribou as a typical domestication story though. More of a co-dependency or symbiotic relationship in such environmental extremes.

Space requirements play right into geography. If you can domesticate and produce the animals food sources, it makes domestication easier. If the geography/topography cannot support feed crops it limits your scale of operation.