We study an Abelian gauge extension of the standard model with fermion families having non-universal gauge charges. The gauge charges and scalar content are chosen in such an anomaly-free way that only the third generation fermions receive Dirac masses via renormalisable couplings with the Higgs boson. Incorporating additional vector like fermions and scalars with appropriate $U(1)$ charges can lead to radiative Dirac masses of first two generations with neutral fermions going in the loop being dark matter candidates. Focusing on radiative muon mass, we constrain the model from the requirement of satisfying muon mass, recently measured muon anomalous magnetic moment by the E989 experiment at Fermilab along with other experimental bounds including the large hadron collider (LHC) limits. The anomalous Higgs coupling to muon is constrained from the LHC measurements of Higgs to dimuon decay. The singlet fermion dark matter phenomenology is discussed showing the importance of both annihilation and coannihilation effects. Incorporating all bounds lead to a constrained parameter space which can be probed at different experiments.
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