The resonance fluorescence properties in the steady-state regime are investigated for a driven cascaded exciton-biexciton quantum dot coupled to the two-dimensional black phosphorus metasurfaces. It is shown that for the material parameters under consideration, both the elliptic and hyperbolic dispersion patterns of the surface plasmon modes are achievable according to the variation of the carrier concentration. Further study on the Purcell factor indicates unequal enhancements in the spontaneous decay of the orthogonal in-plane dipoles. Motivated by this intriguing phenomenon, we then investigate the steady-state properties of the driven quantum dot, where the populations of the dressed levels are highly tunable by engineering the anisotropy of the surfaces. As a result, the manipulation of the carrier concentration will lead to strong modifications in the resonance fluorescence. Under certain conditions, one can observe the squeezing of two-mode noise spectra with different resonances and polarizations. Although at the expense of declines in the photon-sideband detunings, it is feasible to enhance the two-mode squeezing by gate doping. Our proposal can be easily extended to other hybrid systems containing anisotropic metasurfaces, which are important for the development of quantum information science.