What market structure do auto insurance companies operate in? Are they monopolistic competitive?

I will assume you are referring to insurance in the U.S. Insurances of all kinds are definitely competitive in the U.S. The only exception is one or two states. I cannot think of a state that does not have a mandatory liability coverage law, however. So, in that respect, you could possibly, though not likely, state that such and such state is 'monopolistic'; that is, the state allows coverage for auto owners to be provided ONLY by the state or one or more carriers authorized by the state.

If, indeed, there are limited carriers authorized by your state, you can be fairly assured that the non-state carrier has strict limitations as to how to manage claims. ~~ Hope this helps.