In recent times, several hints of lepton flavor universality violation (LFUV) have been observed in semileptonic B decays, which point towards the existence of New Physics beyond the Standard Model. In this context, we consider a new variant of $U(1)_{L_{\mu}-L_{\tau}}$ gauge extension of the Standard Model, containing three additional neutral fermions $N_{e}, N_{\mu}, N_{\tau}$, along with a $(\bar{3},1,1/3)$ scalar leptoquark (SLQ) and an inert scalar doublet, to study the phenomenology of light dark matter, neutrino mass and flavor anomalies. The lightest mass eigenstate of the $N_{\mu}, N_{\tau}$ neutral fermions, plays the role of dark matter. We constrain the new model parameters by using the LFUV observables $R_{K^{(*)}}$ and branching fraction of $b \to s \gamma$ decay processes. We then show the implications of the model on the branching fractions of some rare semileptonic $B$ decay processes with missing energy, e.g., $B \to (K^{(*)}, \phi)+ \not\!{E}$.