  We explore the possible advantages of extending the $\Lambda$CDM model by
more realistic backgrounds compared to its spatially flat RW spacetime
assumption, while preserving the underpinning physics; in particular, by
simultaneously allowing non-zero spatial curvature and anisotropic expansion on
top of it, viz., the An-$o\Lambda$CDM model. This is to test whether the latest
data support spatial flatness and/or isotropic expansion, and, if not, to
explore the roles of spatial curvature and expansion anisotropy (due to its
stiff fluid-like behavior) in addressing some of the cosmological tensions. We
first present the theoretical background and explicit mathematical construction
of An-$o\Lambda$CDM; combining the simplest anisotropic generalizations of the
RW spacetime, viz., the Bianchi type I, V, and IX spacetimes. Then we constrain
this model and its particular cases, viz., An-$\Lambda$CDM, $o\Lambda$CDM, and
$\Lambda$CDM, by using the data sets from different probes, viz., Planck
CMB(+Lens), BAO, SnIa Pantheon, and CC data, and discuss the results.
Ultimately, we conclude that, within the setup under consideration, (i) the
data confirm the spatial flatness and isotropic expansion, though a very small
amount of present-day expansion anisotropy cannot be excluded, e.g.,
$\Omega_{\sigma0}\lesssim10^{-18}$ (95\% C.L.) for An-$\Lambda$CDM from
CMB+Lens, (ii) the introduction of spatial curvature or anisotropic expansion,
or both, on top $\Lambda$CDM does not offer a possible relaxation to the $H_0$
tension, and (iii) the introduction of anisotropic expansion neither affects
the closed space prediction from CMB(+Lens) nor does it improve the drastically
reduced value of $H_0$ led by the closed space. We discuss why it is important
and indispensable to maintain the geometric generalization work program,
especially in models that offer solutions to cosmological tensions. [abridged]
