Mobility is essentially your ability to control a joint movement through a range of motion. It's important to explore and work to improve your mobility limations through exercise. Being able to control movements through a full range of motion allows you to take your muscles from fully extended to fully contracted to get the most out of each exercise. It also allows you to perform the exercise correctly and avoid injury. For example, if you are have good mobility in your shoulders, you may be able to perform a standing shoulder press exercise with good form and without compensating with other parts of your body. If your shoulder mobility is limited, you may find yourself arching your back in order to press weight above your head. In this case we would recommend working on your shoulder mobility while modifying the shoulder press in order to keep your back safe. Perhaps try the landmine press!
One important factor in the mobility equation is control of the movement. If you're only able to reach your hands straight up above your head with the help of a dowel, this might be an indication of flexibility rather than mobility. Keep working some drills with the goal of being able to lift your arms above your head without any assistance. This will likely result in some reduced stiffness and ultimately more gains!