The way that we currently build theories in physics is to write down a classical theory, and then 'quantize it' (which involves replacing classical numbers with operators and enforcing some non-commutation. Or it involves promoting the idea that the action is extremized with a path integral over the action). String theory is no exception, you typically start with a classical string-action.

Because of this, most of the underlying structure of quantum mechanics comes along for the ride. Unfortunately, this usually leads to formal problems (no one has yet developed a satisfying axiomatic quantum field theory, and the situation in string theory is even worse), but physicists ignore these issues, because such theories, while not formally developed, make the right predictions.