  In this work, we derive for the first time observational constraints on the
extended Minimal Theory of Massive Gravity (eMTMG) framework in light of
Planck-CMB data, geometrical measurements from Baryon Acoustic Oscillation
(BAO), Type Ia supernovae from the recent Pantheon+ samples, and also using the
auto and cross-correlations cosmic shear measurements from KIDS-1000 survey.
Given the great freedom of dynamics choice for the theory, we consider an
observationally motivated subclass in which the background evolution of the
Universe goes through a transition from a (positive or negative) value of the
effective cosmological constant to another value. From the statistical point of
view, we did not find evidence of such a transition, i.e. deviation from the
standard $\Lambda$CDM behavior, and from the joint analysis using Planck + BAO
+ Pantheon+ data, we constrain the graviton mass to $< 6.6 \times 10^{-34}$ eV
at 95% CL. We use KIDS-1000 survey data to constrain the evolution of the
scalar perturbations of the model and its limits for the growth of structure
predicted by the eMTMG scenario. In this case, we find small evidence at 95% CL
for a non-zero graviton mass. We interpret and discuss these results in light
of the current tension on the $S_8$ parameter. We conclude that, within the
subclass considered, the current data are only able to impose upper bounds on
the eMTMG dynamics. Given its potentialities beyond the subclass, eMTMG can be
classified as a good candidate for modified gravity, serving as a framework in
which observational data can effectively constrain (or confirm) the graviton
mass and deviations from the standard $\Lambda$CDM behavior.
