We present an interface and geometry preserving method for modeling fully Eulerian multiphase fluid-structure interaction via phase-field formulation. While the time-dependent mobility model preserves the hyperbolic tangent interface profile, the proposed method maintains the geometry of the solid-fluid interface by reducing the volume-conserved mean curvature flow. The reduction of the curvature flow is realized by constructing a gradient-minimizing velocity field (GMV) for the convection of the order parameter. The constructed velocity field extends the solid velocity in the normal direction throughout the diffuse interface region. With this treatment, the GMV reduces the normal velocity difference of the isosurfaces of the order parameter, alleviating the undesired thickening or thinning of the diffuse interface region due to the convection. As a result, the required free energy minimization to regularize the interface profile is significantly reduced, and further, a lesser volume-conserved mean curvature flow from free energy minimization. Using a unified momentum equation via phase-dependent interpolation, we can solve the FSI problems in a fully Eulerian frame of reference, which facilitates the rigid motion and topological changes of interfaces. We first demonstrate the interface preservation effect by rising bubble problems. The geometry preserving effect is shown in the case of a solid block deformed by channel flow. Finally, the ability of the framework is illustrated via the ship-ice interaction problem with contact and complete hydraulic effects.