We discuss a list of possible light gauge boson interpretations for the long-standing experimental anomaly in $(g-2)\mu$ and also recent anomalous excess in $KL \rightarrow \pi^0 + \text{(invisible)}$ events at the J-PARC KOTO experiment. We consider two models: i) $L\mu – L\tau$ gauge boson with heavy vector-like quarks and ii) $(L\mu – L\tau) + \epsilon (B3 – L\tau)$ gauge boson in the presence of right-handed neutrinos. When the light gauge boson has mass close to the neutral pion in order to satisfy the Grossman-Nir bound, the models successfully explain the anomalies simultaneously while satisfying all known experimental constraints. We extensively provide the future prospect of suggested models.