Is it just that Leia’s is sensitive to other female characters who are hurting? Was Leia told by Chewy, Han, or Finn how Rey had really taken to Han as a father figure, and she feels for her/wants to comfort her? Or… does Leia know who Rey really is?
I got the impression that Leia and Han both either knew or had some inkling as to who she was. They both took a shine to her right away. The way they interacted with her was too familiar. They also acted pretty similarly with each other add they did to her. My guess is that this is to show either a connection or some understanding of her.