We investigate the orbital origin of the Fano-Kondo line shapes measured in STM spectroscopy of magnetic adatoms on metal substrates. To this end we calculate the low-bias tunnel spectra of a Co adatom on the (001) and (111) Cu surfaces with our density functional theory-based ab initio transport scheme augmented by local correlations. In order to associate different d orbitals with different Fano line shapes we only correlate individual 3d orbitals instead of the full Co 3d shell. We find that Kondo peaks arising in different d levels indeed give rise to different Fano features in the conductance spectra. Hence, the shape of measured Fano features allows us to draw some conclusions about the orbital responsible for the Kondo resonance, although the actual shape is also influenced by temperature, effective interaction, and charge fluctuations. Comparison with a simplified model shows that line shapes are mostly the result of interference between tunneling paths through the correlated d orbital and the sp-type orbitals on the Co atom. Very importantly, the amplitudes of the Fano features vary strongly among orbitals, with the 3z² orbital featuring by far the largest amplitude due to its strong direct coupling to the s-type conduction electrons.