The problem often isn't the lack of language - we get on just fine with our non-communicative pets after all - but the social capacity of the species involved.
Humans on Earth get on just fine with dogs, cats and other domestic breeds because we have bred them to maximize their social instincts. While some breeds of domestic dogs may resemble wolves, there's a very large difference between canis familiaris and canis lupus. Essentially dogs have become capable of social bonding outside of their species, and so have we.
Our closest ape cousins on the other hand - bonobos and chimps - are more like wild animals. While they form social units among their own species they rarely form social bonds with other species. The same is true of most ape species, which is something else that sets us apart.
Various studies have been done into domestication, including the classic Russian Farm Fox study. While its conclusions have been recently called into doubt (google "domesticated silver fox" for more info), the results are still interesting. While vulpes vulpes kits acted in many ways similar to puppies, as they matured they became solitary and aggressive. After several generations of breeding for passivity they managed to produce a breed of silver fox that apparently retained some of its juvenile characteristics.
All of which becomes relevant to your question when we consider the idea of domestication of cousin species during the story's history.
Imagine if during our early days humans had met and befriended other ape species rather than whatever dog species we originally joined with. From an evolutionary perspective there are a few good reasons for this not to happen, so you're probably going to need a good reason for the genetically distinct (ie non inter-fertile) species to join together. Mutual survival plus a lack of other suitable help-mates is an option here.
Over hundreds of generations of mutual support the companion species would become domesticated and integrated into the society of the dominant species. Their physical similarities might make the bond with them closer than dogs, or it could lead to centuries of abject slavery... humans did it to themselves, no reason they wouldn't do it to their cousins too if the opportunity arose.
In order to reach the point you seem to be interested in there will need to be some great enlightenment period where the other species are finally accepted as being simply less bright citizens. They may not have language, probably won't ever develop it (just like our modern pets won't, without help), but are capable of learning to do reasonably complex tasks. They are naturally social creatures, not just among themselves but with the other species they developed alongside. All held together and guided by the dominant intelligent species.
For a little motivation, the dominant species may have a very low fertility. They can maintain a survival population but need the help of the other species to grow a functional civilization. Fortunately the other species are far more fecund, so there is seldom any concern about labor shortage. As long as you have enough of the dominant species around to coordinate the work, of course.
Where this diverges significantly from cats and dogs is that the partner species in this case is actually capable of evolving language. Their brains are already capable, they already vocalize for various reasons, they can learn non-verbal communications like sign language to a fairly high degree of proficiency, etc. They may not be quite as high on the intellect scale, but physically they're more than a match for the dominants.
Won't be far down the line before you work force starts organizing for better living conditions, fairer wages and all the rights afforded the dominants. Better be prepared for it or it might get bloody.
