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.
As you said, every School is different, but generally it is not unusual that a Professor in Theoretical Physics to belong to a Mathematics school. In my limited experience, though, it is perhaps more common for these faculty members to focus on research (as Theoretical Physics is in fact a very advanced mathematics), rather than on teaching, particularly on undergraduate level.
I have definitely seen folks with PhDs in physics work as lecturers in math departments, where they typically teach things like the calculus sequence, introductory linear algebra, and Boyce & DiPrima differential equations. I think it would be much harder, today, to get a tenure track positions in a math department without a math or statistics PhD, though I can think of a few senior figures in the field whose degrees are in physics and not math.