Consider that animals have sufficient language for the communication they wish to do. It's just not usually formulated verbally in the way that we communicate ( potential exceptions to this include birds and cetaceans ) so we tend to pay it less attention than it warrants.
These signals tend to have evolved and consequently run deeper than our culturally attuned communications so they don't vary across communities in the same way- I could have a conversation with a horse in Australia, the US or here in Europe and make myself understood equally easily by each of them, although of course the language gap is much smaller than the species gap there in any case. However as I mentioned above, they are very limited- restricted to simple suggestions "get out of my space", "move", "groom with me", "danger", "let's run" and so on. Among equines and many other social mammals this communication is predominantly through body language.
For another species to have a spoken language they would consequently need to be capable of a degree of abstract thinking, be equipped with vocal cords or something equivalent and have a need to communicate with their fellows, which implies that social animals would be much more likely to develop language than solitary ones. They would also need an ability to view events in a more future/past way rather than responding simply to the creature's immediate needs.
Interestingly where communication does take place vocally, dialects appear to arise in nature and as languages tend to be a long term consequence of diverging dialect, it is quite plausible that languages will naturally evolve in any vocal populations.
As for whether different animals could communicate between species through a common language, that is an interesting question- would the different lifestyles and experiences of different types of animal mean that any type of communication would be impossible? Wittgenstein suggested that if a lion could speak we would not understand what she had to say.
Effectively, when your animals speak like people, you are well up the road of anthropomorphism, a lot further than most people realise. There isn't a strong real-world grounding for anything along these lines, so I would just pick the system that works best for the story you want to tell.