The anomalous magnetic moment (AMM) of the muon $a_\mu=(g-2)/2$ is one of the most precisely measured quantities in physics. Its experimental value shows, in 2023, a $5.2\, \sigma$ discrepancy $\delta a_\mu=(249 \pm 48) \times 10^{-11}$ with its theoretical value calculated in the standard model framework, using a data-driven ($R$ ratio) dispersive method to calculate the Hadron Vacuum Polarization (HVP) contribution. Meanwhile, lattice QCD numerical calculations of this contribution ($L$) have also yielded a significant discrepancy with respect to the data-driven value ($R$), reaching also $\sim 5\, \sigma$ for the ratio of their best determinations (in reduced windows), $a_{\mu L}^{\rm HVP}/a_{\mu R}^{\rm HVP}=1.0257\pm0.0052$. We suggest here a common solution to these two problems. In standard quantum mechanics, mass ratios and inverse Compton length ratios are identical. This is no longer the case in the special scale-relativity (SSR) framework, in which the Planck length-scale is identified with a lower limit scale, invariant under dilations and replacing the zero point. Consequently a generalized form of Compton relation holds in this theory. Regarding the HVP contribution to the muon $g-2$, the lattice QCD calculation is performed in position {\it space-time} while the data-driven result is calculated in {\it momentum} space. The lattice QCD result is therefore theoretically predicted to be too high by a factor $\rho=1.0287$, which is fully compatible with the observed excess. Once corrected, the lattice and $R$ ratio HVP contributions agree within uncertainties. As regards the muon $g-2$ theoretical calculation, it involves a mass-dependent contribution which comes from two-loop vacuum polarization insertions due to electron-positron pairs and depends on the electron to muon mass ratio $x=m_e/m_\mu$. Using the renormalization group approach, we show that, in this relation, $\ln x$ logarithmic terms depend on mass, while linear $x$ terms are expected to actually depend on inverse Compton lengths. By defining the constant $\Cs_0=\ln(m_\Pl/m_0)$ in terms of the Planck mass $m_\Pl$ and of a reference mass $m_0$, the resulting scale-relativistic correction writes $\delta a_\mu= -\alpha^2 \, (x \:\ln^3 x)/(8 \; \Cs_0^2)$, where $\alpha$ is the fine structure constant. The numerical values of this correction for $m_0=m_\mu$, $\delta a_\mu=(230 \pm 15) \times 10^{-11}$, would fully account for the observed experiment-theory difference.