In the quest for exotic superconducting pairing states, the Rashba effect, which lifts the electron-spin degeneracy as a consequence of strong spin-orbit interaction (SOI) under broken inversion symmetry, has attracted considerable interest. Here, to introduce the Rashba effect into two-dimensional (2D) strongly correlated electron systems, we fabricate non-centrosymmetric (tricolor) superlattices composed of three kinds of $f$-electron compounds with atomic thickness; $d$-wave heavy fermion superconductor CeCoIn$_5$ sandwiched by two different nonmagnetic metals, YbCoIn$_5$ and YbRhIn$_5$. We find that the Rashba SOI induced global inversion symmetry breaking in these tricolor Kondo superlattices leads to profound changes in the superconducting properties of CeCoIn$_5$, which are revealed by unusual temperature and angular dependences of upper critical fields that are in marked contrast with the bulk CeCoIn$_5$ single crystals. We demonstrate that the Rashba effect incorporated into 2D CeCoIn$_5$ block layers is largely tunable by changing the layer thickness. Moreover, the temperature dependence of in-plane upper critical field exhibits an anomalous upturn at low temperatures, which is attributed to a possible emergence of a helical or stripe superconducting phase. Our results demonstrate that the tricolor Kondo superlattices provide a new playground for exploring exotic superconducting states in the strongly correlated 2D electron systems with the Rashba effect.