In the focusing Manakov system, multiple modulation instability (MI) branches coexist on the same plane wave background, so the usual weak periodic modulation cannot selectively excite a single vector Akhmediev breather (AB). Here we propose an eigenvector-based initial perturbation scheme that constructs the initial condition as a plane wave plus Fourier modes whose coefficients follow the perturbation eigenvector of a selected MI branch, enabling controllable high-fidelity excitation of desired vector ABs. Numerical simulations show near-100% fidelity with the exact AB solution. The underlying mechanism is eigenvector-controlled mode selection. The perturbation eigenvectors of the non-Hermitian Bogoliubov-de Gennes operator resolve coexisting MI branches at the identical perturbation frequency, enabling selection of the target MI branch. High-fidelity excitation of the prescribed AB pattern is attained either for comparable gains or when the targeted branch exhibits a sufficient gain advantage. The proposed method provides a simple and robust framework for controllable generation of vector ABs over a broad parameter range, highlighting the key role of eigenvector selectivity in multi-component nonlinear systems.