Task-oriented grasp planning poses complex challenges in modern robotics, requiring the precise determination of the grasping pose of a robotic arm to grasp objects with a high level of manipulability while avoiding hardware constraints, such as joint limits, joint over-speeds, and singularities. This paper introduces a novel manipulability-aware (M-aware) grasp planning and motion control system for seven-degree-of-freedom (7-DoF) redundant dual-arm robots to achieve task-oriented grasping with optimal manipulability. The proposed system consists of two subsystems: (1) M-aware grasp planning; and (2) M-aware motion control. The former predicts task-oriented grasp candidates from an RGB-D image and selects the best grasping pose among the candidates. The latter enables the robot to select an appropriate arm to perform the grasping task while maintaining a high level of manipulability. To achieve this goal, we propose a new manipulability evaluation function to evaluate the manipulability score (M-score) of a given robot arm configuration with respect to a desired grasping pose to ensure safe grasping actions and avoid its joint limits and singularities. Experimental results demonstrate that our system can autonomously detect the graspable areas of a target object, select an appropriate grasping pose, grasp the target with a high level of manipulability, and achieve an average success rate of about 98.6%.