I believe that the butterfly reference in the "butterfly hug" represents resurrection and new life. After you wrap yourself in a "cocoon" embrace, you are calmed and emerge with a "new life" no longer controlled by your mental disorder in a triggered event.
Moon Young tears wings off butterflies in what appears to be a way for her to release her anger by torturing them. Sang Tae is terrified by butterflies flying after him and he connects it with his mother's murder. I do not know what the butterflies represent in these situations or if there is any connection, but it is something to look forward to discovering as the story unfolds.
Moon Young is tortured by her anger and tortures butterflies. (could have another reason)
Sang Tae is terrified and haunted by butterflies.
Kang Tae uses a "butterfly hug" to help heal others.