The name doesn't tell him anything and certainly doesn't tell him what the guy is doing in Batman's Cave. He stares back at him, half glaring, cautious. "I am. I still have no idea who you are and what you're doing here."
He's starting to get insecure which is something he hates feeling. He thought he'd know what to do once in the Cave or in the Manor because it was a place he would know but he doesn't. It's just similar enough to tease him and different enough to makes his frown deepen. And he's looking lost, unable to hide that expression efficiently.
It's pretty empty where he is and if he didn't know he was under the Manor and that the chances for it to be a Cave under the Wayne Manor with a vigilante in it that isn't Batman's is pretty thin. Especially since said vigilante knows Robin.
Dick spots the elevators a bit further - okay, maybe there are several levels of Cave. That'd be something new.