  We address the question of the uniqueness of spatially flat $\Lambda$CDM-like
evolution for FLRW cosmologies in General Relativity, i.e. whether any model
other than the spatially flat $\Lambda$CDM can give rise to the same type of
scale 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, we
consider two models with interaction in the dark sector: coupled fluid-fluid
model and coupled quintessence model. We enforce the \emph{kinematic}
degeneracy with the spatially flat $\Lambda$CDM model via the cosmographic
condition $j=1$ ($j$ being the jerk parameter), which in turn fixes the
function of the interaction term that is a priori unspecified. We argue that in
General Relativity this cosmographic condition is consistent only with spatial
flatness. Employing a dynamical system approach, we show that the spatially
flat coupled fluid-fluid interacting models kinematically degenerate with
$\Lambda$CDM must necessarily be based on a phantom fluid, whereas the set of
physically viable spatially flat coupled quintessence models with power law or
exponential potential kinematically degenerate to $\Lambda$CDM is of measure
zero. Our analysis establishes that coupled fluid-fluid models with non-phantom
fluids or coupled quintessence models with power law and exponential potential
can never reproduce a cosmological evolution similar to that of the
$\Lambda$CDM. The astrophysical consequences of our findings are qualitatively
discussed in light of observational cosmological tensions.
