The ordering of simple fluids in two dimensions was investigated using the residual multiparticle entropy (RMPE) as a measure of the relevance of correlations involving more than two particles in the configurational entropy of the system. To this end, we performed Monte Carlo simulations of two prototype systems, i.e., Lennard-Jones particles and hard discs. Consistent with previous studies, we found that, on approaching the freezing transition, the RMPE of the fluid undergoes a change from negative to positive values. However, in two dimensions the vanishing of the RMPE appears to be more directly related to the formation of six-fold orientationally ordered patches, a process which foreshadows the freezing transition. The specificity of the structural condition attained by the fluid in a state corresponding to a vanishing RMPE was further corroborated by an analysis of the shape of the radial distribution function (RDF): in fact, it turns out that the spatial profiles of the RDF of the Lennard-Jones fluid along a zero-RMPE locus can be superimposed at medium and large distances notwithstanding the difference of density and/or temperature of the corresponding thermodynamic states. The same long-range profile of the RDF is shared also by hard discs in the cited condition. Such a "scaling" property also holds in three dimensions where it provides a suggestive nexus between the ordering criterion based on the vanishing of the RMPE and the Hansen-Verlet freezing rule.