According to conservation of linear momentum and angular momentum, one can derive that Cauchy stress tensor is symmetric and hence has only 6 independent components. Is it possible that, when breaking one or both of conditions, Cauchy stress tensor is not symmetric?
The stress tensor needs not be symmetrical for complex fluids, i.e. fluids with a non-trivial microscopic structure. Nematic liquid crystals are an archetypal example of such materials.
Nematic liquid crystals are complex liquids that possess long-range order in molecular orientation. Think for example about solid rods in water. If the rods are dilute enough their orientation is isotropic and uncorrelated. But when the concentration of the rods increases above a certain critical concentration, the rods start to align and their orientation becomes correlated over macroscopic distances.
The reason why the stress tensor does not need to be symmetrical in this case is because of the local orientation, i.e. because the rod-like molecules have a direction. At equilibrium, all the molecules are aligned in the same direction. In a non-equilibrium situation where the direction of some molecules is not aligned, there must be additional torques trying to re-align these molecules parallel to their equilibrium orientation. These additional torques are exactly the asymmetrical contributions of the stress tensor.
To come back to your question, the symmetry of the Cauchy stress tensor follows from conservation of the angular momentum only if there are no such additional torques.
See e.g.: P.-G. de Gennes, The Physics of Liquid Crystals, 22.214.171.124.
Symmetry of that tensor directly arises from the standard relation of momenta of forces and temporal derivative of angular momentum. So, when assuming standard hypotheses of Newtonian Physics, the stress tensor necessarily turns out to be symmetric. Failure of symmetry is actually equivalent to the failure of the usual interplay of momenta of forces and angular momentum.
The symmetry of the Cauchy stress tensor is the result of applying conservation of angular momentum to an infinitesimal material element. This derivation assumes that there are no body moments.
If there are body moments, the only way that angular momentum can be conserved is if the stress tensor is asymmetric. This is apparently important in the analysis of stress for a polarized dielectric solid under the action of an electric field, materials where the molecular structure is taken into consideration (e.g. bones), solids under the action of an external magnetic field, and the dislocation theory of metals.
For more information, see pages 159-160, 198 of Introduction to Continuum Mechanics by Lai, Rubin, and Krempl.