  Understanding physical mechanisms relies on the accurate determination of
fundamental constants, although inherent limitations in experimental techniques
introduce uncertainties into these measurements. This paper explores the
uncertainties associated with measuring the fine-structure constant ($\alpha$)
and the proton-to-electron mass ratio ($\mu$) using observed fast radio bursts
(FRBs). We select 50 localized FRBs to quantify the effects of varying this
fundamental coupling on the relation between dispersion measure and redshift.
By leveraging independent measurements of dispersion measures and redshifts of
these FRBs, we constrain the uncertainties in $\alpha$ and $\mu$ approximately
to $\Delta\alpha/\alpha=1.99\times 10^{-5}$ and $\Delta\mu/\mu=-1.00\times
10^{-5}$ within the standard $\Lambda$CDM cosmological framework. Remarkably,
these constraints improve nearly an order-of-magnitude when considering a
dynamical dark energy model. This investigation not only yields one of the most
stringent constraints on $\alpha$ and $\mu$ to date but also emphasizes the
criticality of accounting for the energy scale of the system when formulating
constraints on fundamental parameters.
