We investigate the interplay of spin-orbit coupling, electronic correlations, and lattice distortions in the 5d¹ double perovskite Ba₂MgReO₆. Combining density-functional theory (DFT) and dynamical mean-field theory (DMFT), we establish the Mott-insulating character of Ba₂MgReO₆ in both its cubic and tetragonal paramagnetic phases. Despite substantial spin-orbit coupling, its impact on the formation of the insulating state is minimal, consistent with theoretical expectations for d¹ systems. We further characterize the electronic properties of the cubic and tetragonal phases by analyzing spectral functions and local occupations in terms of multipole moments centered on the Re sites. Our results confirm the presence of ferroically ordered z² quadrupoles in addition to the antiferroic x²-y²-type order. We compare two equivalent but complementary descriptions in terms of either effective Re-t2g frontier orbitals or more localized atomic-like Re-d and O-p orbitals. The former maps directly on a physically intuitive picture in terms of nominal d¹ Re cations, while the latter explicitly demonstrates the role of hybridization with the ligands in the spin-orbit splitting and the formation of the charge quadrupoles around the Re sites. Finally, we compare our DFT+DMFT results with a previous DFT+U study of the tetragonal paramagnetic state. We find good qualitative agreement for the dominant charge quadrupoles, but also notable differences in the corresponding spectral functions, underscoring the need for more comparative studies between these two methods.