Kid was prepared to play along with the detective's new quirk up until the other asked for his name. "You... don't know me? Kaitou Kid?"

At that point, Kid was finally willing to consider that perhaps this wasn't just his own confusion. That, or he was really hurt a lot more than just this strange lethargy and being covered in blood.

He must be dreaming. It would explain the mysterious blood on his suit and a yukata wearing tantei asking to be addressed by name.

"Nope, don't remember what happened. Just another heist, then I felt a dark shape coming over me. I stumbled here when I woke up..." He chuckled, "Or is that fell asleep?" No harm in telling some truth to a dream companion. Kid scratched at his neck, flaking away some of the dried blood.