It's allowed because the publishers let it happen. They have every right to stop gameplay videos and streamers, but they don't do it.
For example, Nintendo has issued a ton of DMCA takedowns for "Kaizo Mario" videos on YT. It's within their right to do so. But publishers generally don't issue takedowns because they realize how much Let's Play-style videos and Twitch streaming affects their sales, especially among younger audiences. If a popular streamer starts playing one of their titles, they will see a revenue jump that week... Like during the World of Warcraft fiasco last year, some major WoW streamers switched to various other MMOs and shortly thereafter there was a major surge in Final Fantasy XIV subscriptions to the point where the game sold out of digital copies (because connect queue waits were saturated).
Game streaming (often) encourages consumers to try the game themselves. Movies are a one-and-done kind of experience. Publishers know this and set their enforcement policies accordingly.