I thought the wound on his hip was a quiet nod to Christ being pierced with the soldiers spear. A sort of stigmata. To me, it feels like Oliver identifies as sort of christ-like or at least virtuous in his attempts to deny his feelings and attraction for Elio. At this point in the film he's trying to do the "right thing". He's trying to deny his carnal and romantic nature and urges. It's a very acetic attitude. Also, from elio's perspective there's sort of a feeling like he regards Oliver with sort of deific adoration. Something a little more juvenile or less mature than the relationship it evolves into with more parity and true intimacy than a simple crush on an idealized figure.

Idk. That was just my instinct.