The classic 2D Prince of Persia games, such as Prince of Persia and Prince of Persia 2: The Shadow and the Flame, have their cohesive narrative with recurring characters and settings. These games are linked, with a continuous story arc featuring the same prince protagonist and setting.
In contrast, the newer 3D Prince of Persia games, such as Prince of Persia: The Sands of Time, are a complete reboot of the series. They introduce a new art style, a new prince character, and a new storyline that centers on elements such as time manipulation.
In essence, while there is no single overarching storyline that runs through all Prince of Persia games, the classic 2D titles have their own interconnected narratives. Meanwhile, the 3D games provide a fresh take on the Prince of Persia universe, with a unique protagonist and gameplay mechanics.
If you've got more insights to add, don't hesitate to let us know.