Is playing "Yu-Gi-Oh!" and watching its animated series considered forbidden (haram) due to its inclusion of polytheism (shirk), such as their saying about some cards, "gods," even if the player or viewer does not believe in that?

It is not permissible to play or watch anything that contains forbidden elements. The prohibition becomes more severe if the forbidden element is shirk (polytheism), such as naming other than Allah as deities or imitating polytheists in offering sacrifices to other than Allah. Imitating disbelievers is prohibited, especially regarding matters of creed. Playing the game does not make you a disbeliever. The questioner is suffering from severe waswasa (whispers/obsessive thoughts) and should disregard them.