Is the animagus form connected to the soul(or whatever else the "essence" of a person is)?

What got me thinking about this was the fact that Quirrell is a snake animagus. But we learn from Voldemort that he had almost lost hope at ever having a body again when Quirrell stumbled over one of his horcruxes. So Voldemort jumped at the first opportunity. We know that animagi are rare and the probability of stumbling over a snake animagus is really small. So then is the animagus form anchored to the soul? Or do you have to go through the process of becoming an animagus each time you get a new body, and the soul only determines your form? And I can also imagine that you need both soul and body to turn into an animal, so that the real Quirrel couldn't have changed after Voldemort stopped posessing him.