  We investigate the phenomenology of light GeV-scale fermionic dark matter in
$U(1)_{L_\mu - L_{\tau}}$ gauge extension of the Standard Model. Heavy neutral
fermions alongside with a $S_1(\overline{3}$,$1$,$1/3$) scalar leptoquark and
an inert scalar doublet are added to address the flavor anomalies and light
neutrino mass respectively. The light gauge boson associated with
$U(1)_{L_\mu-L_\tau}$ gauge group mediates dark to visible sector and helps to
obtain the correct relic density. Aided with a colored scalar, we constrain the
new model parameters by using the branching ratios of various $b \to sll$ and
$b \to s \gamma$ decay processes as well as the lepton flavour non-universality
observables $R_{K^{(*)}}$ and then show the implication on the branching ratios
of some rare semileptonic $B \to (K^{(*)}, \phi)+$ missing energy, processes.
