I would argue that the story elements from games and other forms of media, such as books, film and plays, actually do share certain story telling traits, but only at the broadest level of story telling. Cyberdrama, as Janet Murray calls it, is "a collaborative improvisation, partly generated by the author's coding and partly triggered by the actions the interactor takes within the mechanical world." (Murray 5) That is, part of the cyberdrama is created by the creaters of the game itself. I would argue that the authors of the game detail the skeletal structure of the game-story and let the interactors (players) flesh out the details.
Taking the first person shooter Halo as an example, players have no control over the overarching path that the protagonist, Master Chief, takes. The job of creating this path was left to the authors of the game. No matter how you play Halo, (assuming you win), the Halo ring WILL be destroyed and Master Chief will survive. This control of the story parallels the other forms of media. No matter how many times you read Harry Potter or watch Saving Private Ryan, the ends will always be the same. Some may argue that some video games differentiate from this formula by offering multiple endings. I would argue against this by saying that there is (almost) always a "best" ending among those multiple endings, and the player will consider that ending to the be true ending.
The difference, then, between gamestory and other media story is the details of how the protagonist arrives at the final destination. You can't change the fact that Master Chief beats his enemies, but you CAN control how he does it, and the how generally does not alter the course of the story. Losing all of your teammates will yield the same result as saving them all when it comes to the storyline laid out by the authors. Other media story telling on the other hand is not controllable. The protagonist in the other forms of media is developed by how they complete the actions. The details of how a protagonist beats his enemies are essential to the development of the story and of the character.
This difference in story telling leads to different levels of connection between the interactor and the characters. No matter how hard an author tries to describe the protagonist as a good, moral person, interactivity within the game world allows the interactor to partly contradict this identity. When Rockstar Games published Grand Theft Auto: Vice City Stories, they attempted to portray the protagonist as a good man caught in the bad dealings of the city. In the cut scenes, the protagonist seemed to care about the fact that he had just murdered countless thugs. Yet the interactivity associated with the game allows the user to go and murder innocent civilians in the game world after that cut scene is finished, contradicting the personality set out by the authors. The cut scene character and the in game character seem to be two different personalities that can be molded differently. The success of GTA:VCS was more limited than a close counterpart, GTA:LCS partly due to this contradiction of character. In GTA:LCS, the protagonist according to the cut scenes was violent in nature so it was no stretch to see him commit the same random acts of violence in-game.
Even without contradictions of character, I think that, at least in the cases of first person shooters, the connection to the main character is not as strong as in other forms of media. However, I do think that the nature of the game can vary the levels of connection based on the amount of freedom and structure the authors give.
A little ways into this video, the crowd cheers at the return of Jim Raynor, one of the main charactesr from Starcraft 1. The connection here is relatively stronger than other types of videogames and the details about what happens to him are carefully detailed out in the cut scenes. The authors are giving the interactors something to help them connect with the complex character of Jim Raynor.