The National High School Tournament, also know as the Inter High Championship, is the tournament in which compete several high school basketball teams from inumerous japanese prefectures.
To be able to compete on the national level, a team has to be placed as number one or two in said prefecture inter high tournament. For example, Shohoku and Kainan were respectively number two and one in the Kanagawa prefecture, while Aiwa and Meiho were the number two and one in the Aichi prefecture.
There seems to be some error in translation in Kainan's first match during the tournament second round. According to the manga, they win against Umamiya from Iwate prefecture; but said team simply does not exist when we first see the National High School Tournament bracket. One possible explanation would be that the name of the team was wrongly translated to Mamita Nishi and then changed to Umamiya.
Since Hanamichi was the protagonist of the series and got hurt during the game against Sannoh, being unable to play for the rest of the tournament, the manga doesn't focus on continuing the championship, only including some facts like Shohoku losing to Aiwa Gakuen in the third round and Kainan getting the #2 place.
In 2017, an interview with Inoue Takehiko revealed that Kainan was no. 2, but the winning team was not drawn. Inoue also mentioned he did not want a team that most people would deem historically strong to win. The winning team is not Meihou or Daiei Gakuen. Source: https://www.asahi.com/articles/ASK5J5692K5JUTQP022.html
Since the champion team did not appear, fans presumed the team won is "Hakata" since they are in the opposing position of Sannoh in the tournament bracket.
Sannoh is based on 能代工業(秋田県)
Kainan is based on 洛南(京都府)
Aiwa Gakuen is based on 愛知工業大学名電(愛知県)
In 1995, in the real Interhigh, 湘南工科大学附属(神奈川県) (Shonan High, Kanagawa) placed 4th for the first and only time in Interhigh history. Shonan is the model school for both Ryonan and Shohoku. In the same year, Rakunan (Kainan's model school) placed no. 3. Sannoh's model school placed first.
Sannoh's model school has won interhigh for 22 years since it started. It currently holds the record as the basketball school team that has won the most Interhighs, other than Ohka Gakuen for Women (25 wins at Interhigh).