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For the newly discovered $W$-boson mass anomaly, one of the simplest dark matter (DM) models that can account for the anomaly without violating other astrophysical/experimental constraints is the inert two Higgs doublet model, in which the DM mass ($m_{S}$) is found to be within $\sim 54-74$ GeV. In this model, the annihilation of DM via $SS\to b\bar{b}$ and $SS\to WW^{*}$ would produce antiprotons and gamma rays, and may account for the excesses identified previously in both particles. Motivated by this, we re-analyze the AMS-02 antiproton and Fermi-LAT Galactic center gamma-ray data. For the antiproton analysis, the novel treatment is the inclusion of the charge-sign-dependent three-dimensional solar modulation model as constrained by the time-dependent proton data. We find that the excess of antiprotons is more distinct than previous results based on the force-field solar modulation model. The interpretation of this excess as the annihilation of $SS\to WW^{*}$ ($SS\to b\bar{b}$) requires a DM mass of $\sim 40-80$ ($40-60$) GeV and a velocity-averaged cross section of $O(10^{-26})~{\rm cm^3~s^{-1}}$. As for the $γ$-ray data analysis, besides adopting the widely-used spatial template fitting, we employ an orthogonal approach with a data-driven spectral template analysis. The fitting to the GeV $γ$-ray excess yields DM model parameters overlapped with those to fit the antiproton excess via the $WW^{*}$ channel. The consistency of the DM particle properties required to account for the $W$-boson mass anomaly, the GeV antiproton excess, and the GeV $γ$-ray excess suggest a common origin of them.