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We address the question of the uniqueness of spatially flat $\Lambda$CDM-likeevolution for FLRW cosmologies in General Relativity, i.e. whether any modelother than the spatially flat $\Lambda$CDM can give rise to the same type ofscale factor evolution. Firstly, we elaborate on what we exactly imply by a$\Lambda$CDM-like evolution or kinematic/cosmographic degeneracy with the$\Lambda$CDM model, using the lessons from the statefinder diagnostic. Then, weconsider two models with interaction in the dark sector: coupled fluid-fluidmodel and coupled quintessence model. We enforce the \emph{kinematic}degeneracy with the spatially flat $\Lambda$CDM model via the cosmographiccondition $j=1$ ($j$ being the jerk parameter), which in turn fixes thefunction of the interaction term that is a priori unspecified. We argue that inGeneral Relativity this cosmographic condition is consistent only with spatialflatness. Employing a dynamical system approach, we show that the spatiallyflat coupled fluid-fluid interacting models kinematically degenerate with$\Lambda$CDM must necessarily be based on a phantom fluid, whereas the set ofphysically viable spatially flat coupled quintessence models with power law orexponential potential kinematically degenerate to $\Lambda$CDM is of measurezero. Our analysis establishes that coupled fluid-fluid models with non-phantomfluids or coupled quintessence models with power law and exponential potentialcan never reproduce a cosmological evolution similar to that of the$\Lambda$CDM. The astrophysical consequences of our findings are qualitativelydiscussed in light of observational cosmological tensions.

On the uniqueness of $\Lambda$CDM-like evolution for homogeneous and isotropic cosmology in General Relativity