It's an extremely private and clearly emotional moment. If Nanami knew it were safe not to be there at all, he would leave before the box opening stage. Unfortunately, someone has to witness this trying experience, so he does so with the least reaction and judgment to her emotional reactions as possible. As is her due.

The ring inside looks like a diamond. It's clearly an expensive ring and looks like an engagement ring. Nanami is keenly grateful that he did not have to open the box for her and hand her the ring. Small blessings. Her reaction clearly shows she's engaged to someone else (someone not present), so it's truly about that. Not him. He's wallpaper.

"You're welcome, Ari," Nanami says, "This box was clearly labeled for you, not me, so it would have been wrong to steal it. I'm Nanami, from Tokyo." Besides, he was paying it forward, essentially, for the help he received. It's basic human decency. Even Gojo showed it.