When the game takes place.
This is a direct prequel to Luigi's Mansion: Dark Moon, after Mario's already been kidnapped but before E. Gadd contacts Luigi. Luigi is on his way to his mansion where he plans on spending a weekend watching football while leisurely laying on the sofa, belly-button hanging out and everything, when he decides to stop at Princess Peach's castle since it's on the way. Mario is expected to come as well and Luigi, Peach and the two Toads start having tea while reserving Mario's place by putting a spare hat of his on it (incidentally the same hat Luigi found in the washer machine in the first Luigi's Mansion). Just as they're about to dig in the tea and cake, Bowser crashes the party and throws Luigi and the Toads to Acorn Plains, seizing the castle. Since Luigi's mansion is the other way, he needs to trek back to Peach's castle and so decides to defeat Bowser and save the princess while he's at it, hoping that Mario will eventually arrive and take over for him. But it never happens because he's already been summoned to Evershade Valley and kidnapped by the Boos. Incidentally, King Boo's become a little too obsessed with Luigi and started collecting various paraphernalia of him in the form of images, paintings and whatnot while plotting his revenge, but Nabbit stole them all from the various Ghost Houses and scattered them around the levels. The only thing he didn't take is King Boo's favorite piece in the Meringue Clouds Ghost House, because he couldn't find the path to the secret exit where the artwork is located.
Anyway, Luigi saves the princess and finally gets to his mansion, where his lazy weekend is interrupted by E. Gadd summoning him to Evershade Valley. Why Luigi doesn't always jump higher
Any time he was a Moveset Clone of Mario (such as in every previous New Super Mario Bros. game), he would hit a Mario Block offscreen before each stage. Or perhaps a different variety that lasts longer. New Super Luigi U takes place in an alternate universe
This would explain Bowser using the exact same plan from New Super Mario Bros. U, and also how the same spots in the world map have radically different stages for each game. Plus, the several depictions of Luigi throughout the stages seem to indicate he's a revered hero in this universe, as well as Peach's love interest. And while Nabbit is a thief in both universes, in New Super Luigi U he shows a heroic streak that he doesn't seem to have in New Super Mario Bros. U.