Accurately solving the anisotropic interface problem is one of the difficulties in aerospace plasma applications. Based on cubic Cartesian meshes, this paper develops a trilinear nonhomogeneous immersed finite element (IFE) method for solving the complex anisotropic 3D elliptic interface model with nonhomogeneous flux jump. Compared with the existing 3D linear IFE spaces based on tetrahedron meshes, the newly designed trilinear IFE space for the target model simplifies the mesh generation, significantly reduces the number of mesh elements and interface elements, provides much more convenient and efficient ways for finding the intersections between interfaces and mesh edges, and decreases the errors. These advantages lead to much higher efficiency when solving complex anisotropic interface problems in practice. In addition, the proposed method can be easily incorporated into other typical methods based on Cartesian meshes, such as the particle-in-cell method for plasma simulation. Numerical experiments are provided to verify the optimal accuracy, high efficiency, and reliability of the proposed method for solving complex interface problems, as well as its applicability to practical plasma thruster problems.