Organ

Components of an organism. Independent structural units with a specific function. Not the musical kind of organ.
Here it refers specifically to the spiral tube formation within the inner ear. When Shiki said that his organ had broken, he was probably talking about how the ringing in his ears had stopped.
Shiki is something of a poet, don't you think?

Tsukihime Data Collection: Tsukihime Dictionary
§

Components of an organism. Independent structural units with a specific function. Not the musical kind of organ. Here it refers specifically to the spiral tube formation within the inner ear. When Shiki said that his organ had broken during one of the routes, you can interpret that as meaning that the ringing in his ears had stopped.

Tsukihime Dokuhon Plus Period: Tsukihime Dictionary Revised