A Galerkin-free model reduction approach for fluid-structure interaction (FSI) is presented in this article. The reduced order model (ROM) is based on proper orthogonal decomposition (POD), where a reduced basis is formed using energy dominant POD modes. The reduced basis also consists of characteristics POD time modes that are derived from the POD time modes (coefficients) by using their periodicity. In addition to flow variables, the solution state vector comprises the mesh deformation, taking into account the structural deformation in FSI. A ROM solution is obtained by performing a linear interpolation of the reduced basis for changing operating/control parameters. The proposed Galerkin-free POD-ROM approach is demonstrated in terms of two test cases: a canonical case study of vortex-induced vibration (VIV) of a cylinder at Reynolds number Re = 100, where simulations are performed for various structural-to-fluid mass ratios; and a shock wave boundary layer induced panel flutter. For the second case, we use previously computed high-fidelity simulations, considering only the effect of panel thickness on the aeroelastic coupling between the flexible panel and shock wave boundary layer interaction (SWBLI); the inflow is at Mach 2 and Reynolds number based on panel length Rea = 50000. The presented Galerking-free ROM procedure is clean and robust for large mesh deformations, in addition to a significantly lower cost of computation compared to the FSI high-fidelity simulations.