The recent Fermilab measurement of the muon anomalous magnetic moment yields $4.2 sigma$ deviation from the SM prediction, when combined with the BNL E821 experiment results. In the Type-X two Higgs doublet model with the Higgs alignment, we study the consequence of imposing the observed muon $g-2$, along with theoretical stabilities, electroweak oblique parameters, Higgs precision data, and direct searches. For a comprehensive study, we scan the whole parameter space in two scenarios, the normal scenario where $h_{rm SM} = h$ and the inverted scenario where $h_{rm SM}=H$, where $h$ ($H$) is the light (heavy) CP-even Higgs boson. It is found that large $tanbeta$ ($gtrsim 100$) and light pseudoscalar mass $M_A$ are required to explain the muon $g-2$ anomaly. This, in turns, implies that it is difficult to maintain the theoretical stability unless the scalar masses satisfy $M_A^2 simeq M_{H^pm}^2 simeq m_{12}^2 tanbeta approx M_{H/h}^2$. The direct search bounds at the LEP and LHC exclude the small $M_A$ window below $m_{h_{rm SM}}/2$. We also show that the observed electron anomalous magnetic moment is consistent with the model prediction but the lepton flavor universality data in the $tau$ and $Z$ decays are not. For a separate exploration of the model, we propose the golden mode $pp to A h/AH to 4 tau$ at the HL-LHC, which has a high potential to probe the whole surviving parameter space.
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