I think there's some kind of family system, but I'm not sure as I've seen parents refreshing with their children...
However sometimes I hear yukkuri call for little ones.
As far as I can tell, yeah, the biggest catch with families are just specific lines (well, not much else we could have to begin with, I guess?), since they'll either call their ranks ("little one", "daddy", "little sister"). I THINK there were some specific lines but I'm not too sure, I was trying to get some more to breed (all the previous spawn got done with in one playthrough because why else do we play stress relief games?) to be sure if they don't have more specific lines between just calling out for another.