High-precision displacement control for water-hydraulic artificial muscles is challenging because of their strong hysteresis characteristics, which are difficult to be modeled precisely. Recently, data-driven control methods have attracted considerable attention because they do not explicitly use mathematical models, making the design much easier. In our previous work, we proposed a fictitious reference iterative tuning (FRIT)-based model predictive control (FMPC), which combines data-driven and model-based methods for the muscle, and showed its effectiveness because it can also consider input constraints. However, the problem in which control performance strongly depends on prior input-output data remains unsolved. Adaptive FRIT (A-FRIT) based on directional forgetting has also been proposed; however, achieving the desired transient performance is difficult because it cannot consider the input constraints, and there are no design parameters that directly determine the control performance. This paper proposes a novel data-driven adaptive model matching-based controller that combines MPC with the A-FRIT. The experimental results show that the proposed method can significantly improve the control performance and achieve high robustness against inappropriate initial experimental data while considering the input constraints in the design phase.