Kouyou and Yosano never met, or at least an interaction between them was never shown, since Kouyou had been in the Mafia since a young age, and Yosano worked for Mori until she was thirteen. As the Armed Detective Agency and the Port Mafia are rival agencies, it is very likely that both have heard of each other.
Kouyou makes her first appearance in Episode 17, attacking Kyouka and Atsushi. In the midst of the battle, the Guild appears and ambushes both of them. Kouyou is apparently gravely injured, and the Detective Agency takes her hostage. When Dazai went to interrogate her, she was unharmed, so it's likely that Yosano had healed her beforehand. They are also both shown in the background when Pushkin was caught by the Armed Detective Agency and Port Mafia.
As a child, Kouyou tried to escape the Port Mafia with an unknown man she admired, however, they were captured and the man was killed. Yosano also met a man she admired a lot when she worked for Mori, and he died as well under Yosano's hand. This is a parallel people can draw from both women, and Yosano may have turned out the same way as Kouyou if not for Fukuzawa interfering with Mori's plan. They both hold a deep dislike for Mori.
Bungou Stray Dogs Wan has the only animated interactions between Yosano and Kouyou. One in Episode 7 where Yosano-sensei and Kouyou-sensei walk in together to suggest better books for the kids to read, and an ending credit scene where they are strolling along together.
They are very close in age, as Kouyou is 26 and Yosano is 25, and both have eyes the same color as their hair. Moreover, both are women bosses in their agencies, and have strong and sadistic personalities, and both appear to have a big concern for Kyouka
The reason people start shipping them is that they both have similar backgrounds and personalities. Some people believe that Yosano could show the “light” that Kouyou never had access to, because Yosano herself was shown it with the help of Ranpo. During the time Kouyou studied for the Mafia, she was made to believe that because of her powers, she would never have a chance to go to the light.
An extremely popular headcanon for the two is that they are lesbians because of their personalities. The real-life Akiko Yosano was an extremely influential feminist, and it is highly likely that she was queer, due to the poems she wrote on lesbian love and feminism.
Kousano can be considered a rarepair, because neither parties have met each other in canon, though some fans believe that they have met in canon during The Guild arc and Cannibalism arc. The fandom usually writes their interactions in modern-day AUs, or set before/after the current timeline. It is the 14th most written ship in the Bungou Stray Dogs fandom, on AO3. Yosano's rival ship would be Ranposano, while Kouyou has no significant other ship.
|Bungou Stray Dogs Ships • Bungou Stray Dogs Characters|
|SHIPS||het||AkuHigu • Atsukyoka • AtsuLucy • Ranposano|
|slash||ChuuAku • ChuuAtsu • ChuuRan • Dazatsu • Fyolai • Fyoya • KuniChuu • Kunikidazai • Odazai • Ranpoe • Shin Soukoku • Shirachuu • Soukoku • Suegiku|