The statement "Two objects of the same size will always have the same mass" is not correct. While it might seem intuitive that objects of the same size would also weigh the same, mass depends on the material composition of the objects, not just their volume.
Two objects can occupy the same volume (size) but have different densities. Density is defined as mass per unit volume. For example, a cube of wood and a cube of iron can both be the same size, but the wood is significantly less dense than iron. Therefore, the iron cube will have a much greater mass than the wood cube.
In summary, size alone does not determine mass—what matters is the density and composition of the materials involved.