The hairpin or whatever it is that Tsuru has on her hair is Crest of the Kozuki Clan and not only this we can see Kozuki crest in recent chapters many times.
She can’t just carry this crest being a commoner as a commoner would suffer by the hands of shogun for doing such things. I think she is a higher up in Kozuki family. Either she is some hot shot noble or she is from the side family.
One more thing is that Zoro himself might be descendant of Kozuki family line or may from side branch. I have a hunch for this and I will provide some evidence from my side too. First is when Zoro stopped the attack on Tsuru, She might’ve been surprised by attack which can be seen below.
Now, there is a manga by Oda which is DBZ X One Piece cross over in which zoro was shown like this.
Oda in this has hinted that Zoro is also somehow connected to Kozuki Clan.
What do you guys think about this, Please provide your valuable feedback in comments.