On the one hand, Senpai is actually awkward enough to not know her name after spending tons of time with her and her friends.
On the other hand, in Japanese culture actually asking someone for their first name implies that you want to call each other by the first name. And being on first name basis is pretty much exclusive to close friends and family. So to Nagatoro it seems like Senpai wants to confirm that they are close friends.
Yet another beta-blunder blurt out turned miraculously into a glorious Chad-pai move
Also imagine smug Sis-toro when Nagatoro and Senpai leave the room on first name basis