Based on a True Story — There was a now mostly-forgotten Star Trek fanfic that had a similar plot. "Visit to a Strange Planet Revisited" was published in the 1970s in Star Trek: The New Voyages, a collection of fanfics edited by Sondra Marshak, Jacqueline Lichtenberg, and Myrna Culbreath. In the story, William Shatner, Leonard Nimoy, De Forest Kelley, and James Doohan step on the transporter set, and end up on the actualU.S.S. Enterprise in a crisis with the Klingons.
Fake Nationality: Parodied with Arab-American Tony Shalhoub playing Chinese-named Fred Kwan/Sgt. Chen, a reference to Japanese-American George Takei playing totally not Japanese-named Sulu on Star Trek, Englishman Patrick Stewart playing French Jean-Luc Picard on TNG, and Canadian James Doohan playing Scotty, a Scotsman.
Kwan isn't even his real name. He visibly squints for the camera in an attempt to make his eyes look Asian.
Hey, It's That Guy!: Justin Long, the Mac Guy (from the "I'm a Mac, and I'm a PC" commercials) in his first film role as one of the Galaxy Quest fanboys.