We study an unusual effect of spectral-band replication in the optical spectra of dimers, consisting of spherical nanoparticles or nanodisks with a silver core and a J-aggregate shell of TDBC-dye. It consists in the emergence of a doubled number of plexcitonic spectral bands compared to the case of a plasmonic dimer and in narrow peaks associated with the resonances of the J-aggregate shell. The plexcitonic bands can be divided into two groups: the "original" bands, accurately reproducing plasmonic peaks, and their "replicas," with a specific mutual arrangement and intensity distributions. The effect is interpreted using the multi-state effective Hamiltonian model describing a strong coupling between the quasi-degenerate Frenkel excitonic modes in the organic shells and multiple plasmonic modes in the pair of Ag-cores. We quantitatively explain some available experimental data on the optical properties of nanodisks and suggest a way for the observation of the replication effect. Our results extend the understanding of the nature of plexcitonic coupling to more complex systems compared to individual metal/J-aggregate nanoparticles.