In this paper, the electromagnetic scattering from overfilled cavities with inhomogeneous anisotropic media is investigated. To solve the scattering problem, a Petrov-Galerkin finite element interface method on non-body-fitted grids is presented. We reduce the infinite domain of scattering to a bounded domain problem by introducing a transparent boundary condition. The level set function is used to capture complex boundary and interface geometry that is not aligned with the mesh. Non-body-fitted grids allow us to save computational costs during mesh generation and significantly reduce the amount of computer memory required. The solution is built by connecting two linear polynomials across the interfaces to satisfy the jump conditions. The proposed method can handle matrix coefficients produced by permittivity and permeability tensors of anisotropic media. The final linear system is sparse, making it more suitable for most iterative methods. Numerical experiments show that the proposed method has good convergence and realizability. Meanwhile, we discover that the absorbing properties of anisotropic media clearly and positively influence the reduction of radar cross section. It has also been demonstrated that the method can achieve second-order accuracy.