There can be many different parametrizations of the same surface, but they will each have the same topological and geometric properties.

So we could have several correct models--i.e. parametrizations which have the correct characteristics--and a single theory--i.e. set of characteristics with which to choose parametrizations. But each model will predict different yet-undiscovered characteristics. Using these new characteristics to make predictions would constitute different theories which are consistent-as-of-yet. Given the current state of quantum physics, different theories seem to be able to remain consistent for a while.

My use of model and theory have been prescriptive based on my interpretations of the thread below. I didn't intend this but all of the confusion and disagreement seems to have been in our minds.