There was a Netflix show I watched about Dogs which included a bit about a study that would indicate it is a genetic trait which makes a suitable pet. To summarize, it starts with the fact that ALL domestic dogs from Chihuahua to Great Dane share common ancestry - one particular species of Gray Wolf, apparently. Now the interesting thing is the incredible variance in domestic dog variety while the Wolf remains pretty much the same.
The study takes some newborn wolves and attempts to raise them as they would a domestic dog, but finds that after a certain point they can no longer maintain them indoors as they end up acting like wolves despite the upbringing being the same as a domestic puppy. While the dogs tame, the wolves are still "wild" in nature and do not accept the same social restrictions on behavior that the dogs will. I found this surprising, as I always tended to believe in nurture having an effect on nature to a greater degree than this would indicate.
HOWEVER, there was some deeper exploration into this idea which I found really interesting when they discussed an experiment which was begun about 50 years ago with Arctic Foxes. In THAT experiment, they took a sampling of several hundred of members of the species and noted that out of the wild animals, they fell into one of three categories: there were some which were instinctively afraid of people, some which were aggressive towards people, and some which were neither afraid nor aggressive. It was the that third grouping which is described as having the "tameness" gene.
When bred selectively with others from within each grouping, they found the tendencies magnified to some degree - as demonstrated by one of the handlers whose finger is bitten by one of the aggressive foxes right through the cage even though there is no provocation of any kind. Most interesting was that the foxes selected for that lack of fear and lack of aggression were MUCH more easily assimilated to human life and were affectionate and apparently quite excellent companions and remarkably well suited to be pets.
There was talk of perhaps a new choice in these foxes alongside the traditional cats and dogs because they were well adapted for that. And the most fascinating development was that while the aggressive and fearful of the species had not altered their physical attributes in the generations of breeding over these last 50 years, the "tame" group had begun to show all kinds of anomalies in their appearance such as fur color and markings, as well as shortened limbs and wider flatter skulls, some with floppy ears (SOOO CUTE!!) and some with curly tails, and so on... variances much like the differences in the domestic dogs which all come from that same stock of genes. None of the changes showed up in the other groups, but somehow that tame trait allowed for the changes to show up!
There is also a genetic trait which is shared by all species of animals and people concerning the way the central nervous system is "hard wired." (I recently learned about this because I stumbled upon something on the internet which pretty much explained my entire life in the concept of "HSP"s - which is representing "highly sensitive" or "Hyper-sensitive" persons. ) In approximately 15-18% of human and animal populations, there are some which essentially have a built in amplifier of all kinds of stimuli which essentially "turns up the volume" on every sight and sound or sensation experienced, essentially lowering the threshold for the amount of stimuli necessary to have an effect when compared with the rest of the population. In humans, such people are sometimes considered "empathic" in the same fashion as the Star Trek Next Generation character Betazoid Deanna Troi. I think this may also play a part in which individuals make a better companion than others in terms of pets as well. I have noticed that certain of the animals I have had as pets over the years (dog, cat, hamster, parakeet, snake) have exhibited a remarkably attractive personality which sets them apart from others of their kind. My first pet of my own, a hamster, was this way, and i have never seen one since that was like it in the thirty years since I lost her.
Possibly there is an energy that attracts like energy which creates the sort of "luck" I have had in the pets I have had, which now includes the Chins which Mom brought home after an impulsive Craigslist inquiry. They were quite sweet, a blond and a grey, and it turned out that one of them was not a male as we had been told they both were, since one day we were quite surprised to find a couple of mini-me's in the cage with them! The fascinating thing was that the personality of the blond is indistinguishable from the parent as both are very inquisitive and bold and amicable, while the grey, although male (we're pretty sure THIS time, anyway), shares the same slightly more shy personality of his similarly colored mother as well as her affinity for running on the exercise wheel! It has taken MUCH longer to establish the trust with that one despite the exact same conditions as his brother in upbringing. I can only tell the blondes apart by the notch on one ear of the younger one now that they are grown since they act almost identically in personality.
So while nurture does matter, it would seem to matter MORE in the case where an individual is already having a genetic predisposition towards adapting to domestication in the first place, else it will not have an effect on the general nature of a creature if they are not wired genetically in the way that will allow it.