Metamaterials, owing to their exceptional programmable nonlinear mechanical responses, hold great promise for applications in flexible electronics, medical devices, and soft robotics. However, the inherent nonlinearity of microstructural mechanical responses poses a significant challenge to their precise regulation. This paper presents a topology optimization method for the tailored design of multi-material microstructures with target nonlinear mechanical responses. The method incorporates a tangent stiffness constraint, which significantly improves the programmable accuracy of nonlinear mechanical responses. A multi-material interpolation scheme for hyperelastic materials is developed, enhancing the flexibility and diversity of microstructure design. A strain energy interpolation scheme is employed to address numerical instability during nonlinear topology optimization. Moreover, the representative volume element combined with periodic boundary conditions is employed to characterize the nonlinear mechanical behaviors of microstructures. Numerical examples of microstructures under various initial configurations and target nonlinear mechanical responses are provided. The results demonstrate that the proposed method is accurate, robust, and widely applicable in realizing programmable nonlinear mechanical responses.