Not exactly. Batman Begins is one interpretation of Batman's origin story, but it's not the definitive or only 'real' one. Different Batman stories and adaptations offer various takes.
Yes, Batman Begins can be considered a mystery fiction to some extent. It has elements of mystery as Bruce Wayne uncovers secrets and faces hidden threats in Gotham City.
Batman Begins tells the origin story of Bruce Wayne. He witnessed the murder of his parents as a child, which deeply traumatized him. This event led him on a journey to fight crime. He trained with the League of Shadows, learning various combat and stealth skills. But he ultimately rejected their extreme methods. Returning to Gotham, he used his family's wealth and resources to create his Batman persona, with the intention of cleaning up the crime - ridden city.
One idea could be that Batman, in his quest to fight crime in Gotham, discovers some sort of interdimensional rift that leads him to the world of The Avengers. He then has to team up with them to stop a common enemy that is using technology from both universes to cause chaos. For example, Loki could be teaming up with some of Batman's rogues gallery like The Riddler to create complex puzzles and illusions that are both magical and technological in nature.