We investigate experimentally and theoretically the magnetic anisotropy of mechanically rigid polymer composites containing magnetic microparticles arranged in field-induced chain aggregates. The microstructure of composites prepared from silicone rubber with two different magnetic powders is characterized by computed x-ray microtomography, and their magnetization is measured as a function of the inclination angle between the applied magnetic field and the chain direction. The experimental results are quantitatively described using a cascading mean-field approach. At high field strengths, the apparent magnetization exhibits a non-monotonic angular dependence, originating from the rotation of the magnetization vector relative to the applied field. Furthermore, we demonstrate that magnetic anisotropy arising from an anisometric sample shape is equivalent to that induced by an anisotropic chain-like microstructure. The experimentally validated model provides a basis for predicting the magnetic response and torque of structured polymer composites under inclined magnetic fields.