Jack does know Yuuki is a vampire (again, Defence Force) but he doesn't know much about her so he's not immediately making the connection that she's who Zero is talking about. So, being Jack and completely unabashed by social conventions, he asks. "And her name is...?"
He needs to know this stuff. "So you're really different. I did quite a lot of research back while Spike was being a problem, so I'm aware there are a lot of differences between vampires from different worlds, but damn, it's hard to keep track of which is which!" A rueful grin, to try to avoid dragging the mood down too far. "Thank you for telling me. I know it can't be easy to just hand over your weaknesses to someone you never met before."