OK, so if you're like me and have no bloody clue what an "oshi" is, it's kinda like "favorite" except with more emotional ties. Imagine there's a particular tennis player you like and always watch their matches and keep up with their public lives - that's your "oshi." It can be translated as "one you support" or "one you push forward," apparently.
So in the game this world is based on, she was his favorite character and he cared more about what happened to her than what happened to the other protagonists.
Anyone who actually speaks Japanese feel free to correct me.