We propose a two-loop induced Zee-Babu type neutrino mass model at the TeV scale. Although there is no dark matter candidate in the original Zee-Babu model, that is contained in our model by introducing an unbroken discrete $Z_2$ symmetry. The discrepancy between the experimental value of the muon anomalous magnetic moment (muon $g-2$) and its prediction in the standard model can be explained by contributions from additional vector-like charged-leptons which are necessary to give non-zero neutrino masses. The mass of vector-like leptons to be slightly above 300 GeV is favored and allowed from the muon $g-2$ and the current LHC data. We find that from the structure of neutrino mass matrix, doubly-charged scalar bosons in our model can mainly decay into the same-sign and same-flavour dilepton plus missing transverse momentum. By measuring an excess of these events at the LHC, our model can be distinguished from the other models including doubly-charged scalar bosons.
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