The duo have never canonically interacted, being from two separate games. However, they do interact in the Ultimate Talent Development Plan bonus mode.
In their event, Kyoko can be seen walking through the halls of the classrooms to view the exhibits. Kazuichi comes up to Kyoko asking if she'd like to join him in the haunted house exhibit. She declines, causing Kazuichi insisting for her to join him. Despite his different ways on suggesting for her to join him, she declines.
However, it is worthy to note Kyoko is very ressemblant to Sonia Nevermind and Kazuichi doesn't seem to mind being around Kyoko Kirigiri implying they are at least friends. Kyoko and Kazuichi also have strong family issues with their fathers and fans usually connected them more to their past then their outward appearance.
The two are complete opposites but it is the reason most people seem to ship it. As the trope goes, Kyoko is a stoic quiet character while Kazuichi is a energetic fun-loving character.
The ship sailed as fans knew about Kyoko and Kazuichis past. Since some fans love the dynamic of, stoic x energetic ships, it was no surprise when this ship caught their eye. Some fans also think their pasts are similar which could lead them to getting along.
However, being a ship between characters from different casts who only interacted once in Ultimate Talent Development Plan, it's a relatively rare pairing. It loses in popularity to both characters' main ships, such as Naegiri or Soudam. The ship is overshadowed, but people still acknowledge it.
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