Why Yugami-kun looks so perplexed after "Rio-Chan"? ?
this is the most belated reply of all time. but I got it:
it's a well known trope / manga knowledge that calling a friend by their main name instead of family name (Fujisawa Rio; Rio is the "first name" while "Fujisawa" is the family name) means that you share some sort of a closer bond. That is, if you don't know her too well, you'd call her Fujisawa. If you consider her someone very close to you or your bff, you'd call her Rio.
So when Rio thinks about what she WANTS and then ends up telling Chihiro that she wants to be called "Rio", the subtext is that her final wish is to have a CLOSE BEST FRIEND that she appreciates a lot. And then they walk off happy together.
This, being her "main wish" or goal in life is perplexing to Yugami. As we all know, by this point in the story, Yugami still sticks to his ideals and believes friendships are worth nothing and should never matter at all in someone's life. So that's why Yugami is just horrifically confused.
edit:
I acknowledge some of the earlier comments have already kinda vaguely answered this one in my stead in slightly different ways, but hey, I'm just gonna leave my thoughts up!