  We investigated the Cu/Ni-O$_2$ bilayer multi-orbital Hubbard model with
Cu/Ni impurity approximation embedded in the O lattice by incorporating the
3d$^{8}$ multiplet structure coupled to a full O-2p band. For this simplified
model of describing the local electronic structure of hole doped homo-bilayer
Cu(Ni)O$_2$-Cu(Ni)O$_2$ and hetero-bilayer CuO$_2$-NiO$_2$, we demonstrate that
the homo-bilayer system hosts rich phase diagram with inversion symmetry
breaking between two layers at strong enough interlayer hybridization,
especially the coexistence of two-hole singlet and triplet states in opposite
layers. Besides, we show that the interlayer hybridization in hetero-bilayer
systems destabilizes the Zhang-Rice singlets or 3d$^{8}$ triplets within each
layer via charge transfer between layers. The relevance of these electronic
redistribution to the bilayer or interface systems of cuprates and/or
nickelates are discussed.
