Understanding and mastering the light-light and light-matter interactions in coupled structures have become significant subjects, as they provide versatile tools for manipulating light in both classical and quantum regimes. Mimicking quantum interference effects in pure photonic nanostructures, from weak Fano dip to intense electromagnetically induced transparency, usually requires strong asymmetries in complex geometries and larger interactions between resonances, i.e., in the intermediate coupling regime. Here, we numerically demonstrate a simple and chemically feasible plasmonic nanocube-hexagonal-nanoplate heterodimer with a strong, tunable self-induced transparency window created by the intermediate coupling between the near-degenerate dark and bright hybridized modes. Further assisted by the strong coupling introduced by the J-aggregate excitons covering the heterodimer, three evident exciton-induced transparency windows were observed. These multiband transparencies in a single-particle-level subwavelength configuration, could on one hand enrich the toolbox of multi-frequency light filtering, slowing and switching beyond the diffraction limit, and on the other hand, work as a fundamental testbed for investigating multiscale light-matter interactions at the nanoscale.