We analyze the process of photon-pair generation via spontaneous parametric down-conversion in a quadratic nonlinear asymmetric waveguide coupler. The two waveguides have different geometry, such that light coupling only occurs within a narrow bandwidth of one of the generated (signal) photon modes, while the other (idler) photon together with the pump stay localized in one (driven) arm of the coupler. We demonstrate that such a setup represents a powerful and flexible tool for engineering spectral properties of generated photon pairs. Mode hybridization and dispersion of coupling can be utilized for shifting the balance between group velocities of interacting pump, signal, and idler fields, subsequently leading to a significant increase of spectral factorizability (purity) of photons. We also show that for interaction lengths shorter than one beat length, generated pairs with signal photon being localized in the auxiliary (not pumped) arm of the coupler appear to be spectrally localized in both signal and idler components. The bandwidth of such intrinsic filtering of generated photons can be controlled by several geometrical parameters.