The balance between convergence and diversity is a key issue of evolutionary multi-objective optimization. The recently proposed stable matching-based selection provides a new perspective to handle this balance under the framework of decomposition multi-objective optimization. In particular, the stable matching between subproblems and solutions, which achieves an equilibrium between their mutual preferences, implicitly strikes a balance between the convergence and diversity. Nevertheless, the original stable matching model has a high risk of matching a solution with a unfavorable subproblem which finally leads to an imbalanced selection result. In this paper, we propose an adaptive two-level stable matching-based selection for decomposition multi-objective optimization. Specifically, borrowing the idea of stable matching with incomplete lists, we match each solution with one of its favorite subproblems by restricting the length of its preference list during the first-level stable matching. During the second-level stable matching, the remaining subproblems are thereafter matched with their favorite solutions according to the classic stable matching model. In particular, we develop an adaptive mechanism to automatically set the length of preference list for each solution according to its local competitiveness. The performance of our proposed method is validated and compared with several state-of-the-art evolutionary multi-objective optimization algorithms on 62 benchmark problem instances. Empirical results fully demonstrate the competitive performance of our proposed method on problems with complicated Pareto sets and those with more than three objectives.