Well, through controlled breeding people have managed to isolate certain characteristics in dogs to the point that even though they are the same species they look significantly different.
However they all still have the same basic biology: four legs, same general shape, a single head with two eyes, a nose, a snout, etc. In Monsters Inc the situation differs significantly.
Some monsters are based on lizards, others on insects, or mammals. Not only do they come in various shapes and sizes, they also vary in the number of limbs they have, the powers they demonstrate (invisibility, flight, etc.), and other critical characteristics which typically define a "species".
So I suggest we take a step back and allow all these different creatures to actually be separate species. However, let them be actual animals. Monsters.
And let the intelligent species be parasitic in nature, inhabiting the body of these monsters and forming a society in which each individual wears the body of a monster the same way that we might wear a suit.
Basically, think along the lines of what the alien invaders in "The Host" are.