  We clarify the relation between noncommutative spacetimes and multifractional
geometries, two quantum-gravity-related approaches where the fundamental
description of spacetime is not given by a classical smooth geometry. Despite
their different conceptual premises and mathematical formalisms, both research
programs allow for the spacetime dimension to vary with the probed scale. This
feature and other similarities led to ask whether there is a duality between
these two independent proposals. In the absence of curvature and comparing the
symmetries of both position and momentum space, we show that $\kappa$-Minkowski
spacetime and the commutative multifractional theory with $q$-derivatives are
physically inequivalent but they admit several contact points that allow one to
describe certain aspects of $\kappa$-Minkowski noncommutative geometry as a
multifractional theory and vice versa. Contrary to previous literature, this
result holds without assuming any specific measure for $\kappa$-Minkowski. More
generally, no well-defined $\star$-product can be constructed from the
$q$-theory, although the latter does admit a natural noncommutative extension
with a given deformed Poincar\'e algebra. A similar no-go theorem may be valid
for all multiscale theories with factorizable measures. Turning gravity on, we
write the algebras of gravitational first-class constraints in the
multifractional theories with $q$- and weighted derivatives and discuss their
differences with respect to the deformed algebras of $\kappa$-Minkowski
spacetime and of loop quantum gravity.
