A class of supernovae (SNe) called SN Ic-CSM are characterized by late-time emergence of narrow emission lines of elements formed in the oxygen core of a massive star. A popular scenario is the interaction of the SN ejecta and O-rich circumstellar medium (CSM), i.e., circumstellar interaction (CSI). Uncovering the progenitor system of SNe Ic-CSM plays a critical role in understanding the final evolution of a massive star to a bare C+O star. In this letter, we present observations of SN 2022esa, which we show is an SN Ic-CSM. Surprisingly, a stable periodicity of ~32 d is found in its light-curve evolution with a hint of a slowly increasing period over ~200 d. We argue that the main power source is likely the interaction of the SN ejecta and O-rich CSM, while the energy input by the post-SN eccentric binary interaction within the SN ejecta is another possibility. In either case, we propose a massive Wolf–Rayet (WR) star as the progenitor, in a WR–WR or WR–BH (black hole) binary that will eventually evolve to a BH–BH binary. Specifically, in the CSI scenario, the progenitor system is an eccentric binary system with an orbital period of about a year, leading to the observed periodicity through the modulation in the CSM density structure. We also show that some other objects, superluminous SN I 2018ibb (a pair-instability SN candidate) and peculiar SN Ic 2022jli (the first example showing stable periodic modulation), show observational similarities to SNe Ic-CSM and may be categorized as SN Ic-CSM variants. Complemented with a large diversity in their light-curve evolution, we propose that SNe Ic-CSM (potentially linked to SNe Ibn/Icn) are a mixture of multiple channels that cover a range of properties in the progenitor star, the binary companion, and the binary orbit.