Why are the commutators for spin taken as postulates?

The commutators for spin taken as postulates because spin itself is not a classical thing. Nevertheless, there is a crude analogy between the spin and the proper rotation of objects in classical motion. And since the rotation in the classical case is described by the angular momentum, it is reasonable to simply assume that this can be similar to describe the spin. When we made this assumption, i.e. postulated, it still needs to be compared with the experiment. If we have agreement, then our assumptions are correctly guessed, but only guessed, and not strictly inferred from more fundamental assumptions.