A cobalt(II) complex and manganese(II) coordination polymer, formulated as [Co2(H2btca)(mbpy)4][H2btca]·4H2O (1) and {Mn2(btca)(mbpy)2(H2O)2}n (2) (H4btca = 1,2,4,5-benzenetetracarboxylic acid; mbpy = 4,4'-dimethyl-2,2'-bipyridyl), constructed by mixed bipyridine-tetracarboxylic ligands were synthesized and characterized. Single-crystal structural analyses reveal that compound 1 is a discrete neutral binuclear molecule, while compound 2 is a two-dimensional (2D) coordination polymer. The metal ions in these compounds are well isolated, with an intramolecular Co2+⋯Co2+ distance of 9.170 Å for 1 and Mn2+⋯Mn2+ separation of 10.984 and 11.164 Å for 2 due to the bulk tetracarboxylic linker. This isolation gives rise to a single-ion magnetism origin of the compounds. Magnetic studies reveal a large zero-field splitting parameter D of 82.6 cm-1 for 1, while a very small D of 0.42 cm-1 was observed for 2. Interestingly, dynamic ac magnetic measurements exhibited slow magnetic relaxation under the external dc field of the two compounds, revealing the field-supported single-ion magnet (SIM) of 1 and 2. The detailed theoretical calculations were further applied to understand the electronic structures, magnetic anisotropy, and relaxation dynamics in 1 and 2. Combined with our recently reported compound (Eur. J. Inorg. Chem., 2022, e202200354), the foregoing results provide not only a rare binuclear cobalt(II) SIM and the first 2D manganese(II) SIM coordination polymer but also a bipyridine-tetracarboxylic ligand approach toward novel SIMs.