Why is the test for sodium as dihydrogen antimonate so specific?
It has no (direct) relation to the electrochemical series nor to acido-basic properties.
It is not a true chemical reaction, rather just crystallization of the least soluble salt from the available ions.
There are no distinguished molecules $\ce{NaCl}$, $\ce{KH2SbO4}$, $\ce{NaH2SbO4}$, $\ce{KCl}$ in the solution, but mutually independent hydrated ions $\ce{Na+}$,  $\ce{Cl-}$, $\ce{K+}$, $\ce{H2SbO4-}$ instead (plus some minor hydrolysis products). Which salt precipitates first depends on their solubility products. $\ce{NaH2SbO4}$ has smaller solubility product than $\ce{KH2SbO4}$.
Similar things would happen if you mix solutions of potassium  chloride and sodium nitrate ( chlorate, perchlorate ). If original solutions are concentrated enough, and especially if you cool the mixture below ambient temperature, there will precipitate the respective potassium salt with the oxoanion.
Another case is the $\ce{NaCl}$ solution  with dissolved $\ce{NH3}$ And $\ce{ CO2}$, with ions $\ce{Na+, NH4+, Cl^-, HCO3-}$, leading to precipitation of $\ce{NaHCO3}$ and solution of $\ce{NH4Cl}$. It is used commercially in the Solvay process
producing sodium carbonate from sodium chloride.