In this work, we show that various quantum interaction radial potential models used for describing diatomic molecules are particular cases of an exactly solvable multiparameter exponential-type potential. This feature is exploited to find the conditions that potentials should fulfill to be considered as equivalent. As an example of the usefulness of the proposal, the obtained requirements are applied to some of the most common diatomic potential models. Specifically, we have shown the equivalence among the radial potentials of Manning–Rosen, Deng–Fang, Schiöberg, Badawi–Bessis–Bessis, Tietz, Wie, Sun, the negative of the Williams–Poulios and the Möbius square potentials. The proposal to obtain equivalent radial potentials is general and can be directly applied to other interaction models as well as in the study of relativistic molecular physics.