We consider incompressible generalized Newtonian fluids in two space dimensions perturbed by an additive Gaussian noise. The velocity field of such a fluid obeys a stochastic partial differential equation with fully nonlinear drift due to the dependence of viscosity on the shear rate. In particular, we assume that the extra stress tensor is of power law type, i.\,e. a polynomial of degree $p-1$, $p \in (1,2)$, i.\,e. the shear thinning case. We prove that the associated Kolmogorov operator $K$ admits at least one infinitesimally invariant measure $μ$ satisfying certain exponential moment estimates. Moreover, $K$ is $L^2$-unique w.\,r.\,t. $μ$ provided $p \in (p^\ast,2)$, where $p^\ast$ is the second root of $p^3 - 8p^2 + 14p -6 =0$, approximately $p^\ast \approx 1.60407$.