We study the aggregation of charged plate-like colloids, Na-montmorillonite clays, in the presence of ionenes, oppositely charged polymer chains. The choice of the charged polymer allows tuning its linear charge density to match/mismatch the average charge separation on the clay surfaces. We assess the nanoscale structure of the aggregates formed by small-angle X-ray and neutron scattering. The nanoscale features of the formed clay aggregates are dominated by the presence of a stacking peak, giving clear evidence for the formation of clay tactoids, that is, a face-to-face aggregation geometry of the clay platelets. The chain charge density of ionenes influences not only the stacking repeat distance within the clay tactoids but also the extent of stacking and abundance of the tactoids. We may distinguish two regimes as a function of clay and ionene polymer charge densities (ρc and ρp, respectively). The first regime applies to ρp > ρc and ρp ≈ ρc, that is, for highly and "matching" charged chains. Under these conditions, the intercalated chains lie in a flat conformation within the tactoids, irrespective of the ionic strength (within the range studied, i.e., up to 0.05 M NaBr). For weakly charged chains, ρp < ρc, undulation of the ionene chains within the tactoid is seen. The degree of undulation increases with ionic strength due to the decreasing persistence length of the ionene chains. The extent of stacking (5-10 platelets per tactoid) is a general feature of all the systems, and its origin remains unknown. The system corresponding to the closest match in charge separations on the clay surface and on the polymer chain (ρp ≈ ρc) features the highest abundance of tactoids. This coincides with the highest macroscopic density as deduced from simple visual inspection of sediment volumes. This leads to the open question regarding the link between the density at the nanoscale and the macroscopic density and sedimentation behavior of the aggregate.