Well, 'The Name Game' in American Horror Story could be a variety of things. It might be a form of entertainment within the horror - filled world of the show that takes a sinister turn. It could also be a mechanism through which the show explores themes like identity and power. For instance, if a character loses the 'name game', they might lose their sense of self or become vulnerable to the evil forces at play in the story.
It adds an element of mystery. Just the idea of a 'name game' makes you wonder what's going on. It can make the atmosphere more tense as the characters participate in it.
An American horror film was called The Shining. The film was released in 1980 and was adapted from Stephen King's novel of the same name by Stanley Kubrick. The film was about a story that gradually involved ghosts and other horror elements. The film received widespread praise and became a classic horror film.