We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebio et al. (J. Fluid Mech., vol. 781, 2015, pp. 298-329) expanding the range of Prandtl number $Pr$ examined by two orders of magnitude from 0.7 up to 70. Significant effects of $Pr$ on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin-Obukhov (M-O) similarity theory and van Driest damping functions. We then employ M-O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number $Rig$ is determined by the length scale ratio $h/L$, where $h$ is the half-channel gap depth and $L$ is the Obukhov length scale. As $h/L$ approaches very large values, $Rig$ asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number $Reb$ $\equiv$ $\epsilon$/($\nu$$N2$), where $\epsilon$ is the dissipation, $\nu$ is the kinematic viscosity and $N$ is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio $L$+ $\equiv$ $L$/$\delta$, where $\delta$ is the near-wall viscous scale. The flux Richardson number $Rif$ $\equiv$ -$B$/$P$, where $B$ is the buoyancy flux and $P$ is the shear production, is found to be proportional to $Rig$. This then leads to a turbulent Prandtl number $Prt$ $\equiv$ $\nu t$/$\kappa t$ of order unity, where $\nu t$ and $\kappa t$ are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number $Frh$ $\equiv$ $\epsilon$/($NU\prime$$2$), where $U\prime$ is a turbulent horizontal velocity scale, is found to vary like $Rig$$\mathrm{<mrow\; data-mjx-texclass="ORD">-1<mrow\; data-mjx-texclass="ORD">/</mrow>2</mrow>}$. All these scalings are consistent with our numerical data and appear to be independent of $Pr$. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn, J. Phys. Oceanogr., vol. 10, 1980, pp. 83-89), together with M-O scalings, results in a parameterization of $\kappa t$/$\nu$ ~ $\nu t$/$\nu$ ~ $Reb$$Rig$/(1-$Rig$). With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M-O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M-O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.