Yui was once Empress of Yamatai. If she wasn't a Taisho, maybe people would still call her Empress the way former U.S. Presidents are still called presidents. But she's not a queen and definitely not a princess. Furthermore, I don't really consider the "princess" title to really exist under (constitutional) Yamatai and these days it makes me (and Hanako) cringe a little every time someone calls Hanako a princess. After all, she's been a servant to the clan as much as she's been "royalty." Remember that Hanako never chose the name Ketsurui, Yui did, and if you listen her her talk about herself, she never calls herself a Ketsurui. She only refers to herself as "Shosho Hanako." But yeah, Yui is not a princess, at least not since the mid YE 20s.