[It has nothing to do with anything, it's true, but the mention of it and Barriss' reaction gives Leia what she needs to know. She's seen Mirialans before, certainly, but non-humans from their universe were rare here, so there was no telling for sure whether Barriss was from the same galaxy, or if she were a different type of being from a different world.]

It doesn't. But if you recognize the name, you and I are from the same galaxy. That's not something that can be said about most of the other beings here.