  The $3.1\sigma$ $R_K$ anomaly after Moriond 2021 and $3.3 \sigma$ $\Delta
a_\mu$ from Fermilab Muon g-2 experiment implicate that the lepton flavor
universality violation (LFUV) may play a role in the exploration of new
physics. A Flavor Gauged Two-Higgs Doublet Model (FG2HDM) is proposed and
investigated in this work. To get rid of the the redundancy in Yukawa coupling
of 2HDM-III, a specific U(1) flavor symmetry is introduced. The charge
difference between two scalar doublet forbid the appearance of pseudoscalar and
hence there are only three particles (a charged and neutral heavy scalar
together with a neutral gauge boson) adding to SM particle spectrum. The heavy
neutral scalar-mediated flavor-changing interactions occur among down-type
quarks. With obvious difference from 2HDM-II, the charged Higgs in FG2HDM can
naturally explain $R_{D^{(*)}}$ anomaly. The heavy neutral vector boson $Z'$,
changing flavor for down-type quark uniquely as well, provides a solution to
$R_{K^{(*)}}$. The anomalous magnetic dipole moment (AMDM) of muon and
electron, especially the new released $\Delta a_\mu$, can further discriminate
$Z'$ parameter space.
