Rhenium disulfide belongs to group VII transition metal dichalcogenides (TMDs) with attractive properties such as exceptionally high refractive index and remarkable oscillator strength, large in-plane birefringence, and good chemical stability. Unlike most other TMDs, the peculiar optical properties of rhenium disulfide persist from bulk to the monolayer, making this material potentially suitable for applications in optical devices. In this work, we demonstrate with unprecedented clarity the strong coupling between cavity modes and excited states, which results in a strong polariton interaction, showing the interest of these materials as a solid-state counterpart of Rydberg atomic systems. Moreover, we definitively clarify the nature of important spectral features, shedding light on some controversial aspects or incomplete interpretations and demonstrating that their origin is due to the interesting combination of the very high refractive index and the large oscillator strength expressed by these TMDs.