  We investigate the interplay of spin-orbit coupling, electronic correlations,
and lattice distortions in the $5d^1$ double perovskite Ba$_2$MgReO$_6$.
Combining density-functional theory (DFT) and dynamical mean-field theory
(DMFT), we establish the Mott-insulating character of Ba$_2$MgReO$_6$ 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 $d^1$ 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 $z^2$ quadrupoles in addition to the antiferroic
$x^2-y^2$-type order. We compare two equivalent but complementary descriptions
in terms of either effective Re-\ttg 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 $d^1$ 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.
