As synthesis of graphene on copper became one of the primary preparation methods for both fundamental research and industrial application, Raman spectra of graphene/Cu systems need to be quantitatively understood regarding how their interactions affect the electronic structure of graphene. Using multi-wavelength Raman spectroscopy, we investigated three types of graphene bound on Cu: graphene grown on Cu foils and Cu film/SiO2, and Cu-evaporated exfoliated graphene. 2D peak frequencies of the first two samples were ~17 cm-1 higher than expected for 1.96 eV excitation even when the effect of strain was considered. More notably, the upshift in 2D decreased with increasing excitation energy. Based on control experiments using Cu-evaporated graphene, we revealed that the spectral anomaly was induced by environment-dependent nonlinear dispersion in the electronic bands of graphene and determined the degree of the electronic modification. We also showed that the large upshifts of G and 2D peaks originating from differential thermal expansion of Cu could be significantly reduced by backing Cu films with dielectric substrates of insignificant thermal expansion. The quantitative analysis of electronic coupling between graphene and Cu presented in this study will be highly useful in characterizing as-grown graphene and possibly in other forms.