Do we have postulates of statistical mechanics the way we have for quantum mechanics?
As usual, when dealing with postulates, there is no reason to believe that the choice is unique. Therefore, taking into account that other formulations are possible, I would consider a fundamental postulate that
"An isolated system in equilibrium is equally likely to be in any of its accessible microscopic states."
This is an often cited postulate at the basis of the so-called microcanonical ensemble. However, I would add another essential postulate necessary to make contact with Thermodynamics;
"Equilibrium Thermodynamics can be obtained from Statistical Mechanics by taking the so-called thermodynamic limit if it exists."
Notice that this requirement allows us to justify the equivalence of different ensembles, thus making it possible to substitute the first postulate with whatever probability distribution is the most appropriate for non-isolated systems.