What is the physical meaning of the Maxwell Stress tensor symmetry?
No, it's not symmetric. Let me explain:
Say for instance that you only take the magnetic part of the Maxwell stress tensor (let's ignore the electric part). Then you would have the outer product $BB$ + (diagonal tensor). A lot of textbooks usually write it as 1/$\mu BB+$ (diagonal tensor), which is wrong and misleading, since it assumes that the material has a linear behavior $B = \mu * H $.
The right expression is $BH+$ (diagonal tensor), where $B = \mu_0 (M + H)$ Therefore if M is not colinear with $H$ you will get a non-symmetric tensor. However if M is colinear with H then you will get a symmetric one. This colinearity between M and $H$ holds true when the magnetization can be described by a linear and isotropic relationship ... that is $M = $some_physical_constant * $H$.