If mathematics is a language used to formally define and establish the relationship between things with out any ambiguity and if physical models are mathematical models used to model observed reality then why there are so many interpretations of those models? Why do they arise? Does different interpretations mean that the model is incomplete or what does it mean? I am just not able to understand relationship between interpretations, mathematics and physical models.

What one chooses to be axiomatic and what one chooses to be derived is a bit arbitrary. Example: the axiom of choice, Zermelo's well-ordering theorem, and Zorn's lemma are equivalent to one another. Assume any one and you can derive the other two. Another example, this time in physics: Lorentz ether theory (LET) and special relativity are mathematically equivalent. The two differ only in their axioms. So which to choose? In the case of ZFC, it's rather arbitrary. In the case of LET vs SR, the axioms of special relativity are simple while for LET it's more a question of WTF.