TanAo

Canon
Aoi and Tanjiro first met when Tanjiro went to the Butterfly Estate in order to heal and train with Zenitsu and Inosuke. Tanjiro is extremely grateful for Aoi's help and care for him and his friends and constantly complements her skills. She appreciates this and at the time, seems to respect Tanjiro for his hard work and how he didn't give up as quickly compared to Inosuke and Zenitsu.

Tanjiro also helps intervene for her when she was getting kidnapped by Tengen and he also helps defend her when Tengen calls her weak and useless by saying that everyone has their own own strengths and circumstances. Aoi is extremely grateful and touched by this gesture that she views Tanjiro as a good ally and friend.

Right when Tanjiro, Zenitsu, and Inosuke are about to leave the Butterfly Estate he runs up to Aoi in order to thank her for all of her hard work. She says that it's no problem and that she was simply doing her job but Tanjiro insists that she's really talented when it comes to training and healing. Aoi brings up that she's glad that she's good at something to which Tanjiro asks her why she said it like that. She then goes on to tell him that she got through final selections by pure luck and can only use water breathing to help kill small minor demons and that she was to scared to even set foot on battle when the time came. She then continues saying that if it weren't for Shinobu and the other girls at the Butterfly Estate then she would've been completely useless. Tanjiro then tells her that that's not true and that she is really talented and if it weren't for her then he wouldn't have become a stronger person and he says that he'll take her wishes out onto the battlefield with him. He then thanks her one more time and the runs away leaving her shocked at the abrupt kindness.

Fanon
The ship sailed as a result of Tanjiro's kindness towards Aoi and how Aoi was grateful for that kindness. Due to the lack of female characters (Besides Nezuko who's Tanjiro's sister) a lot of people thought that Aoi could be a potential love interest. However that theory was later debunked after Kanao was introduced and there was more ship tease between those two instead.

While it's not the most popular shipping choice for either character naturally losing popularity towards ships like Zentan and Inotan for Tanjiro and AoIno and Aoikano for Aoi. And it's not to popular in Fanart or Fanfiction there's still a small portion of the fanbase that ships this and feels as if they could form a good romantic relationship due to the fact that they are on fairly good terms in Canon.