The oblique electromagnetic scattering by a dielectric elliptical cylinder which is coated eccentrically by a nonconfocal dielectric elliptical cylinder is examined in this work. The problem is solved using the separation of variables in terms of Mathieu functions, in combination with the addition theorem for Mathieu functions, a complicated procedure due to the following three factors: the nonexistence of the orthogonality relations for Mathieu functions due to the different constitutive parameters between the two cylinders and the background medium, the complex expressions due to the oblique incidence that leads to hybrid waves for both the scattered and induced fields, and the use of the addition theorem, which introduces a cross relation between even and odd terms. The method described here is exact, its solution is validated compared with other published results from the literature, and the high accuracy is revealed. Both polarizations are examined and numerical results are given for the scattering cross sections, including lossless and lossy materials.