Shadow is a magical construct based in Mai's soul gem, designed to help Mai deal with trauma. She fully manifests when Mai unlocks her signature in ch. 37.
All MGs, indeed all magic users eventually develop a shadow. “Magic is creation fueled by magical energy. Emotions can create magical energy. Live long enough, bury enough subconscious desires and feelings, and eventually enough is fed into the creation of a magical construct to ask the questions you refuse to directly answer. For a natural magic user, this process can lead to… complications. Thankfully, due to the Zenith’s Soul Gem, there are safeguards to help prevent any accidents from happening.” (Shadow, ch 37.)
I suspect that all shadows are named Shadow, but since each is tied to a particular MG, this should not be a problem.
An extensive explanation of Shadow is given in chapter 37. We earlier saw Shadow, without explanations, in chapters 3, 14, and 20. See Mai's dreams for a description of these earlier encounters.