I think you need to ask yourself a question: Do you want them to think like human beings? If their mindset is human, then not only will they be driving off, starving out and/or outright killing other sentient species (the Neanderthals), they'll also be doing the same to other humans and to dumb animals. After all, the real world is full (empty?) of endangered species which are endangered for the simple reason that we humans decided that chopping down their forest and ploughing up their prairie to feed us and our domestic animals is A Good Thing.
If you invent a new psychology for them, the world is your oyster.
In evolutionary terms, you can perhaps get a symbiosis/mutualism between a couple of species in situations where lack of food is not the main cause of mortality. If your aardwolves and aardvarks are under constant threat of being eaten by hyenas, they may cooperate in mutual defence. There are a while range of species which pay attention to each other's alarm calls (peacocks, langur monkeys and chital deer being a famous example). You could escalate this to a group defence from the hyenas.
Also consider that in nature, animals are very rarely eating exactly the same food, in exactly the same place at exactly the same time. It may look the same when we throw it into a broad category ('meat' for instance), or a broad habitat ('the African plains'), but niche separation often means that the reality is more complex. The diet of lions, leopards and cheetahs has some overlap (they'll all kill a young zebra), but only a mental cheetah takes on an African buffalo and only a desperate lion tries to catch a hare. The niche separation is not complete in those cats, so lions will happily kill or drive off leopards and cheetahs.
So perhaps your aardvarks are eating mainly termites and beetle grubs, and your aardwolves are eating mainly ants and caterpillars. They'll both eat all of them, but the aardvark is better at digging into termite mounds and the aardwolf is better at resisting ant stings. For other sentient species, perhaps some catch squid and others catch fish. Perhaps some catch little fish and some catch big fish.
Or there could be a resource sharing... scavenging which developed into something else. Polar bears, for instance eat pretty much only the blubber and brains of the seals they kill. Eating the meat causes them problems with eliminating the nitrogen from the protein (they'd need to drink water, and the only water is ice, which will cost them too much heat). Arctic foxes & seabirds follow polar bears around, hoping to scavenge the meat from their kills. If the fox/bird starts leading the bear to seals, that's the start of a mutually beneficial partnership.