Ben Jonson's Volpone is absolutely based on well-known animal fables, such as those found in Aesop's Fables. Most of the characters are named after animals who share their characteristics. For instance, Volpone is crafty and is accordingly named after a fox. Additionally, the general structure of the story can be seen as analogous to an animal fable about a fox who fakes his death to catch greedy birds. As such, the play certainly relies upon elements of animal fables to discuss the state of human morality.
However, Volpone is a little different from more conventional animal fables: while most animal fables use real animals to comment on human behavior, Jonson uses actual humans, but names them after animals. Thus, the effect is a little different than the conventional animal fable. At the end of the day, Jonson still uses real humans to explore real human activities, appetites, and vices, and so Volpone is not quite the same as a strict animal fable that takes place entirely within the animal kingdom. As such, I think it would be most accurate to say that, though Volpone certainly relies heavily on the animal fable tradition, it should not be viewed in the same sense as a conventional animal fable.
According to wikipedia an animal fable or beast fable is defined as:
a type of fable, in which human behaviour and weaknesses are subject to scrutiny, by reflection into the animal kingdom.
If we look at the fable Volpone in regards to whether or not it's an animal fable, we would have to agree that it is an animal fable. It's an animal fable because animals are talking and scruitinizing human behavior.
Throughout the play the characters are analyzing human behavior. Mos the parasite says:
Bane to thy wolvish nature.
This is referring to Volpone.