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The quest for new physics beyond the Standard Model is boosted by therecently observed deviation in the anomalous magnetic moments of muon andelectron from their respective theoretical prediction. In the present work, wehave proposed a suitable extension of the minimal $L_{\mu}-L_{\tau}$ model toaddress these two experimental results as the minimal model is unable toprovide any realistic solution. In our model, a new Yukawa interactioninvolving first generation of leptons, a singlet vector like fermion($\chi^{\pm}$) and a scalar (either an SU(2)$_{L}$ doublet $\Phi^\prime_2$ or acomplex singlet $\Phi^\prime_4$) provides the additional one loop contributionto $a_{e}$ only on top of the usual contribution coming from the$L_{\mu}-L_{\tau}$ gauge boson ($Z_{\mu\tau}$) to both electron and muon. Thejudicious choice of $L_{\mu}-L_{\tau}$ charges to these new fields results in astrongly interacting scalar dark matter in $\mathcal{O}({\rm MeV})$ range aftertaking into account the bounds from relic density, unitarity and selfinteraction. The freeze-out dynamics of dark matter is greatly influenced by$3\rightarrow2$ scatterings while the kinetic equilibrium with the SM bath isensured by $2\rightarrow2$ scatterings with neutrinos where $Z_{\mu\tau}$ playsa pivotal role. The detection of dark matter is possible directly throughscatterings with nuclei mediated by the SM $Z$ bosons. Moreover, our proposedmodel can also be tested in the upcoming $e^+e^-$ colliders by searchingopposite sign di-electron and missing energy signal i.e. $e^{+} e^{-}\rightarrow \chi^{+} \chi^{-} \rightarrow e^{+} e^{-} \cancel{E}_T$ at thefinal state.

$(g-2)_{e,\,\mu}$ and strongly interacting dark matter with collider implications