Written in its Lagrangian-mean formulation, the Craik-Leibovich (CL) equation with steady Stokes drift is mathematically identical to the governing equation of flow under background rotation. The vertical shear of Stokes drift exerts an effect on Langmuir turbulence in shallow water (SWLT) analogously to the effect of spanwise background rotation on rotating plane Couette flow (RPCF). Here we conduct large eddy simulations to study different flow regimes in SWLT in comparison with RPCF. Analogously to the flow regime transition from organized roll cells to small-scale turbulence in RPCF as the spanwise rotation rate increases, the flow structure in SWLT transits from coherent Langmuir cells to small-scale Langmuir turbulence as Stokes drift shear increases. And eventually enhanced mixing saturates the CL instability so that the flow enters a state of marginal instability. These distinct flow regimes have an important consequence on the vertical mixing of horizontal momentum in SWLT, homogenizing Lagrangian-mean momentum in the former regime and Eulerian-mean momentum in the latter. While this regime transition is mainly governed by the inverse Rossby number \(\mathrm{Ro}^{-1}\) defined from the constant spanwise rotation rate in RPCF, it is affected by both the surface magnitude and decay length scale of Stokes drift in SWLT so that a bulk estimate of the Stokes drift shear given by \(\mathrm{La}_t^{-2}\) (where \(\mathrm{La}_t\) is the turbulent Langmuir number) is not sufficient. We show that Stokes drift shear controls the flow regime transition in SWLT by affecting the turbulence anisotropy and thereby modulating the turbulence-induced anti-Stokes flow.