EscapeMitsuru got a motorcycle to use during operations and nightly trips to Tartarus. Unlike other vehicles, it is fully operational during the Dark Hour. However, Mitsuru did not get one solely for Shadow-related purposes. She very much enjoys riding it as well. To her, the motorcycle was something she could use to escape from her normal routine. She tells the protagonist, "With a motorcycle, I can go where I want, when I want, all by myself. It's my modest attempt at rebelling against my restricted life." Between schoolwork, the Student Council, SEES and family obligations, Mitsuru is extremely busy. She even tells the protagonist it is rare for her to have some free time. In order to function under so much responsibility, I would imagine her day is very structured and scheduled. Her life is almost completely controlled by her family name--right down to the clothes she wears and expectations for her future. Going out and riding her motorcycle, alone, wherever she feels like going is her way of forcing some independence into her life. She can leave everything behind and forget about her stress and responsibility while experiencing the thrill of riding.
Based on the things Mitsuru says about her motorcycle, it is pretty clear that riding was an escape mechanism for her. Due to high expectations and being forced to grow up very quickly, Mitsuru likely did not develop many proactive coping skills. Further, she initially does not have any close relationships and is under the impression that sharing her feelings and concerns will only burden others. With stressors piling every day from her various responsibilities and no one to talk to or support her, it would only be a matter of time before she broke if she had no means of coping with stress. Avoidance, while not proactive or truly helpful, is a way of temporarily reducing anxiety. Because Mitsuru says she no longer has to think of her motorcycle as a way to run from her feelings, I assume she went out for rides when she started to become overwhelmed with emotions. Riding probably took her mind off of everything and she could pretend she was free for the duration of the ride. Her anxiety would then be reduced and she could go back to her life of barely functioning. It's no surprise, then, that as Mitsuru becomes closer to Yukari and the protagonist, she feels as though she no longer needs to use it to run away. She has people who care about her that she can talk to when she is feeling overwhelmed or stressed.
Why did the character developers pick a motorcycle as opposed to another means of transportation for Mitsuru? Perhaps they considered what motorcycles symbolize or represent. They are strongly related to a desire for freedom and need for independence and autonomy. Further, when they appear in dreams, they are said to symbolize wanting an escape from some situation or responsibility in life. Motorcycles are also associated with drive and motivation as well as restlessness. All of these things parallel the reasons why Mitsuru rides her motorcycle. She has a desire for freedom from responsibility, autonomy in her life and restlessness from her overwhelming emotions she is trying to keep down. Mitsuru has incredible motivation and drive to adhere to her responsibilities, despite not wanting to comply completely. In last rank of her social link, however, Mitsuru gives the protagonist her motorcycle key. She ensures him/her she no longer needs to use it as an escape from her feelings. She would face the future head-on and not run away from her responsibilities in life. Riding her motorcycle therefore transforms from an escape mechanism to a form of bonding with a person she really cares about. One last association of motorcycles in dreams is a means by which you move forward in life. I would thus argue the transformation of the motorcycle's meaning to her is very representative of her moving forward in life as well.