We experimentally studied the process of supercontinuum generation in a birefringent multi-core microstructured optical fiber. By selecting the excitation of the fundamental mode, or by combining the first and the second order modes of a particular core, it was possible to emphasize the role of four-wave mixing on the transfer of power from the infrared to the visible region of the spectrum. We carried out an in-depth analysis of the effects of input light polarization on the generated supercontinuum spectral features. The measured polarization properties of the output Stokes and anti-Stokes bands confirmed the strong vector nature of the four-wave mixing processes. The experimental spectra exhibit excellent agreement with numerical simulations of the nonlinear mode interactions.