I get the impression he's trying to confirm something. Kind of like when he asked to see Mikuru's breasts to see if she had that star shaped mole and confirm she wasn't a fake.
Although I have no idea what distinctive trait real Nagato's underwear might carry.
Nagato showing that much emotion? Yeah, that's far out of the ordinary.
Judging for Kyon's coat, I think it's the Disappearance movie. I recall that the only time he's alone in the club room with Nagato while wearing the coat. Could be wrong though. My memory is bad.
