"game identifier" in old versions of GameMaker is an unique number (between 0 and 4294967295) that determines registry section that the game's high scores are going to be stored in. You generally want this to be unique per game - e.g. if you make a second game based on your existing project file, you'd want to change the identifier so that scores don't clash.
In newer versions of GM it is only occasionally used to avoid clashes (e.g. as prefix when storing data on HTML5, where by default all games on same domain may share saved data)