The ground-state spin arrangement and the bipartite entanglement within Cu2+-Cu2+ dimers across the magnetization process of the 4f-3d heterometallic coordination polymer [Dy(hfac)2(CH3OH)2Cu(dmg)(Hdmg)2]n (H2dmg = dimethylglyoxime, Hhfac = 1,1,1,5,5,5-hexafluoropentane-2,4-dione) are theoretically examined using the symmetric isotropic spin-1/2 Ising–Heisenberg orthogonal-dimer chain. The numerical results point to five possible ground states of the compound with three different degrees of the quantum entanglement within Cu2+-Cu2+. Besides the standard ferrimagnetic and saturated phases without quantum entanglement of Cu2+ ions, which are manifested in low-temperature magnetization curve as wide plateaus at the non-saturated magnetization 16.26μB and at the saturation value 20.82μB, respectively, one also finds an intriguing singlet-like phase with just partial entanglement within Cu2+-Cu2+ and two singlet phases with fully entangled Cu2+-Cu2+ dimers. The former quantum phase can be identified in the low-temperature magnetization process as very narrow intermediate plateau at the magnetization 9.27μB per unit cell, while the latter ones as zero magnetization plateau and intermediate plateau at the magnetization 18.54μB. Non-monotonous temperature variations of the concurrence, through which the entanglement within cooper dimers is quantified, point to the possible temporary thermal activation of the entangled states of Cu2+-Cu2+ also above non-entangled ferrimagnetic and saturated phases.