The comprehensive depiction of the many-body effects governing nanoconfined electrolytes is an essential step for the conception of nanofluidic devices with optimized performance. By incorporating self-consistently multivalent charges into the Poisson-Boltzmann equation dressed by a background monovalent salt, we investigate the impact of strong-coupling electrostatics on the nanofluidic transport of electrolyte mixtures. We find that the experimentally observed negative streaming currents in anionic nanochannels originate from the collective effect of Cl- attraction by the interfacially adsorbed multivalent cations and the no-slip layer reducing the hydrodynamic contribution of these cations to the net current. The like-charge current condition emerging from this collective mechanism is shown to be the reversal of the average potential within the no-slip zone. Applying the formalism to surface-coated membrane nanoslits located in the giant dielectric permittivity regime, we reveal a new type of streaming current activated by attractive polarization forces. Under the effect of these forces, multivalent ions added to the KCl solution set a charge separation and generate a counterion current between the neutral slit walls where the pure KCl conductivity vanishes. The adjustability of the current characteristics solely via the valency and amount of the added multivalent ions identifies the underlying process as a promising mechanism for nanofluidic ion separation purposes.