This work investigates the supramolecular aggregation of Cu(II) complexes based on a combination of the heterocyclic N-donor ligand 2-aminopyrimidine (2-apym) and a variety of different carboxylate ligands, like salicylic acid (H₂SAL), maleic acid (H₂MAL), and glycine (HGLY). Four new Cu(II) complexes [Cu(HSAL)₂(2-apym)₂] (I), [Cu(HSAL)₂(2-apym)₂]·(H₂SAL)₂(2-apym) (I′), {(H2-apym)[Cu(GLY)(μ-Cl)₂Cl]}_n (II) and [Cu(μ-MAL)(2-apym)]_n (III) have been synthesized and characterized by elemental analyses, FT-IR spectra, and X-ray structural analyses. I and I′ appear as molecular clusters whereas II is a 1-D coordination polymer with weak axial Cu–Cl interactions. However the last one shows a 2-D coordination polymer with trigonal bipyramidal geometry for Cu(II) ion. The two last cases have no molecular fragments packed with non-covalent interactions. In these new supramolecular compounds, existent species join together with the cooperation of multiple inter/intra-molecular classical O–H⋯O/N, N–H⋯O/N, and non-classical N–H⋯Cl hydrogen bonds (H-bonds), offset face to face π⋯π, edge to face C/N–H⋯π, and lp⋯π stacking interactions in the form of various homo/hetero-synthons leading to architecturally different structures. DFT calculations were used to estimate the binding energy of the involved non-covalent interactions and whole stabilization energy of related network of I, I′, and II. Theoretical calculations facilitate the comparison of intermolecular interactions, which demonstrate that for all of I–II, N–H⋯N and N–H⋯O H-bonds govern the network formation. The equilibrium constants for the three proton-transfer systems including SAL/2-apym, GLY/2-apym, MAL/2-apym, the stoichiometry and stability of complexation of these systems with Cu²⁺ ion in aqueous solution were investigated by potentiometric pH titration method. The stoichiometries of the most complex species in solution was compared to the crystalline Cu²⁺ ion complexes with the cited proton-transfer systems.