  We study an extension of the minimal gauged $L_{\mu}-L_{\tau}$ model in order
to explain the anomalous magnetic moments of muon and electron simultaneously.
Presence of an additional scalar doublet $\eta$ and an in-built $Z_2$ symmetry
under which the right handed singlet fermions and $\eta$ are odd, leads to
light neutrino mass in scotogenic fashion along with a stable dark matter
candidate. In spite of the possibility of having positive and negative
contributions to $(g-2)$ from vector boson and charged scalar loops
respectively, the minimal scotogenic $L_{\mu}-L_{\tau}$ model can not explain
muon and electron $(g-2)$ simultaneously while being consistent with other
experimental bounds. We then extend the model with a vector like lepton doublet
which not only leads to a chirally enhanced negative contribution to electron
$(g-2)$ but also leads to the popular singlet-doublet fermion dark matter
scenario. With this extension, the model can explain both electron and muon
$(g-2)$ while being consistent with neutrino mass, dark matter and other direct
search bounds. The model remains predictive at high energy experiments like
collider as well as low energy experiments looking for charged lepton flavour
violation, dark photon searches, in addition to future $(g-2)$ measurements.
