The Higurashi When They Cry series uses games as character beats and atmosphere. Identifiable games:
No original fictional game is central (unlike Yu-Gi-Oh or Kaiji). Real games ground the setting in 1983 rural Japan.
To play them: Hanafuda decks (Nintendo/Oishi Tengudo), Go sets (Kiseido, online: OGS, Fox), Mahjong (Mahjong Soul, Tenhou), Daifugō (standard deck + rules).