Can somebody with a PhD in theoretical physics become a Assistant Professor / Instructor in Mathematics? Most community colleges and universities ask for PhD / Master's in Mathematics in order to become a Assistant Professor / Instructor in mathematics. I know that this will greatly depend on the particular school, but do people in academia view theoretical physicist (someone with a PhD in theoretical physics) as equivalent to someone who has a PhD in mathematics? It's not mathematical physics, but I would expect theoretical physics to be a mathematical science too.
Can somebody with a PhD in physics become a mathematics professor? Yes. My department has (at least) four such people.

Is this common? No, not particularly. I think my department is rather unusual in that respect.

do people in academia view theoretical physicist (someone with a PhD in theoretical physics) as equivalent to someone who has a PhD in mathematics? No, as a general rule I don't think anybody will say that they're equivalent - theoretical physics is certainly very distinct from math, and most theoretical physicists will not be seriously considered for a position at most math departments. Nonetheless, some theoretical physicists publish work that is close enough to pure math, or even outright do a career change after their PhD and start publishing exclusively in math journals, to make them a good fit for a position in a math department.