A new kinetic model is proposed to solve the Riemann problem for stiffened gas interface from piecewise linear discontinuous initial distributions. In the model gas particles on each side of the fluid interface are reflected back from the interface, respectively, which is moving with a velocity to achieve the force balance between both sides. Compared with the existing Riemann solver, the present model can keep second order accuracy in both space and time. It is also capable of eliminating the numerical mixing at the fluid interface which is different from existing kinetic models. The new model is applied to the numerical flux calculation at a cell interface and with the help of homogeneous equilibrium mixture assumption within a cell, a new gas-kinetic scheme is developed for multimaterial flow with stiffened equation of state. The scheme is tested with several typical high-speed multifluid flows, including the water-gas shock tube flows and the shock-water cylinder interaction. The computed results are in good agreement with other numerical and experimental studies. Fluid interfaces as well as shock waves are sharply captured, free of numerical oscillation near the interface, even for density ratio up to about one thousand, which validate its high accuracy, strong robustness and good parallel performance.