We study the statistics of velocity circulation in two-dimensional classical and quantum turbulence. We perform numerical simulations of the incompressible Navier-Stokes and the Gross-Pitaevskii (GP) equations for the direct and inverse cascades. Our GP simulations display clear energy spectra compatible with the double cascade theory of two-dimensional classical turbulence. In the inverse cascade, we found that circulation intermittency in quantum turbulence is the same as in classical turbulence. We compare GP data to Navier-Stokes simulations and experimental data from [Zhu et al. Phys. Rev. Lett. 130, 214001(2023)]. In the direct cascade, classical and quantum turbulence circulation statistics coincide at low but strongly differ at high orders. We associate this difference with the presence of quantized vortices, which makes enstrophy ill-defined mathematically. Our results establish the boundaries of the equivalence between two-dimensional classical and quantum turbulence.