Spin-torque oscillators (STOs) are devices that allow for the excitation of a variety of magnetodynamical modes at the nanoscale. Depending on both external conditions and intrinsic magnetic properties, STOs can exhibit regimes of mode hopping and even mode coexistence. Whereas mode hopping has been extensively studied in STOs patterned as nanopillars, coexistence has been only recently observed for localized modes in nanocontact STOs (NC-STOs), where the current is confined to flow through a NC fabricated on an extended pseudo spin valve. By means of electrical characterization and a multimode STO theory, we investigate the physical origin of the mode-coupling mechanisms favoring coexistence. Two coupling mechanisms are identified: (i) magnon-mediated scattering and (ii) intermode interactions. These mechanisms can be physically disentangled by fabricating devices where the NCs have an elliptical cross section. The generation power and linewidth from such devices are found to be in good qualitative agreement with the theoretical predictions, as well as provide evidence of the dominant mode-coupling mechanisms.