Controlling the stability of anisotropic particles is key to the development of advanced materials. Here, we report an investigation, by means of mesoscale molecular dynamics simulations, of the stability and structural change of calcium-rich dispersions containing negatively charged nanoplatelets, neutralized by calcium counterions, in the presence of either comb copolymers composed of anionic backbones with attached neutral side chains or anionic-neutral linear block copolymers. In agreement with experimental observations, small stacks of platelets (tactoids) are formed, which are greatly stabilized in the presence of copolymers. In the absence of polymers, tactoids will grow and aggregate strongly due to large attractive Ca2+-Ca2+ correlation forces. Unlike comb copolymers which only adsorb on the external surfaces, block copolymers are found to intercalate between the platelets. The present results show that the stabilization is the result of a free energy barrier induced by the excluded volume of hydrophilic chains, while the intercalation is due to bridging forces. More generally, the results shed new light on the recent finding of the first hybrid cementitious mesocrystal.