@amstash
Kochou was able to reincarnate because she exhausted her powers from reviving Shirogane before her dream part became bigger than her soul.
Decades later, she was reborn as a half-Ayakashi daughter of Sango and Reincarnation!Misaki (the main characters in the Fish arc). It turns out Yuu's mother (from Rain arc) and Kochou's uncle Aki got married, and Yuu the Rainboy was also reborn as their son (iirc he said he'll be reborn as his mom's son again). Reincarnation!Misaki was Yuu's daughter and Aki's granddaughter. So basically Kochou was reincarnated as a descendant of the people she helped. I won't be surprised if she's also related to Saku's girlfriend and Sachie (from the Friend arc) in some way.
The author didn't give a specific reason why Kochou remembered. Maybe Shima simply wanted for the two to become happy, or maybe Shima left hints about the explanation.
I think we can use is Yuu the Rainboy as an example. When Yuu was reincarnated, he received a new name that was similar to his old name (Yuuya) so he had some memories from his past life and was an incomplete Ayakashi. Maybe Reincarnation!Kochou has a similar name too, and I think that's very plausible. Like, either Sango noticed that Kochou was reborn as his daughter, or he simply wanted to show gratitude to Kochou for giving him the push to move on by naming their daughter after her and later on found out that their daughter really is Kochou's reincarnation. The only difference is that Reincarnation!Kochou is half-Ayakashi (and implied quarter-god if Menou really is Kukurihime) so she didn't become an incomplete Ayakashi like Yuu. Anyway, that's just my theory if it makes sense.