Interfacial hydrodynamic instabilities in multicomponent superfluids provide a versatile platform to explore nonequilibrium quantum dynamics beyond classical fluid analogues. We study dynamical interfacial instabilities in a quasi-two-dimensional three-component Bose-Einstein condensate confined in a harmonic trap, where differential effective rotation between neighboring components is generated through selective phase imprinting. A component with an imprinted vorticity $l_j$ acquires an azimuthal phase winding ${\varphi }_j=l_j\theta$, producing a superfluid velocity $v_{\theta }(r_j)=l_j\hslash /m{r}_j$ at interface radius $r_j$ and an effective angular velocity ${\Omega }_j^{eff}=l_j\hslash /m{r}_j^2$. The differential effective rotation $\Delta {\Omega }_j^{eff}={\Omega }_j^{eff}-{\Omega }_{j+1}^{eff}$ between neighboring components constitutes the sole driver of interfacial shear and the Kelvin-Helmholtz instability. Three instability regimes are examined: KHI in the strongly immiscible limit, counter-superflow instability in the partially miscible regime, and a parameter window where both mechanisms coexist. A hydrodynamic pressure-balance model is derived, showing that the KHI onset condition depends only on $\Delta {\Omega }_j^{eff}$. Time-dependent Gross-Pitaevskii simulations and a Bogoliubov-de Gennes analysis identify the dominant unstable modes, whose spatial structures are consistent with the observed density modulations. The dominant azimuthal mode is set by a competition between shear-driven destabilization and surface-tension stabilization, confirmed by numerical simulations with different imprinted vorticities. The results show that a three-component condensate with two tunable interfaces provides a controlled platform for studying multicomponent quantum hydrodynamics, with instability dynamics significantly richer than those of binary mixtures.