The events of the game actually took place and the dream is just Mario's mind recreating them with his memories.
Wart was a dream demon attempting to conquer the Mushroom Kingdom by trapping its citizens in a permanent nightmare. The world of Sub-con is an area of the Dreaming that he cordoned off for this purpose (
as Brute and Gob once did to trap Hector Hall). The scene at the end with Mario waking up was merely showing him breaking free from the dream after he defeated Wart.
Mario wasn't dreaming in SMB 2...
He, and the other characters, were engaging in
Astral Projection. It wouldn't be the weirdest power he's had. Subcon can't be reached by physical beings, so they could only go as souls or some such. Going with the "Peach is a Mage" guess below, it could be her that's keeping them anchored in Subcon. The hit points, lives, and continues represent her ability to keep them there. If her item-creation powers are much more powerful in this form, it would explain why Starmen are much more common and Super Mushrooms can stack for more HP.
Super Mario Bros. 2 was not a dream at all!
Subcon is an alternate universe created by dreams, but Mario and his friends actually did visit the land physically. Wart is, in fact, the dream counterpart of Bowser. The ending of the game that showed Mario asleep did not indicate that his adventure was a dream. He is actually dreaming about the good times he had during one of his finest adventures.
- Alternatively, the game was a dream, but at the same time their adventures in Subcon really did happen. Subcon is real, but the only way to get there is to dream of it, and that Mario and co. all had the same dream together. They really did save a dream world from an evil frog. The Japanese-only BS Super Mario USA supports this theory.
Shy Guys appear in Super Mario Bros. 2 because they're part of Mario's dreams/nightmares.
Remember that SMB2 is revealed to be All Just a Dream at the end. It's common for people to chronically dream about early childhood experiences, especially traumatic ones. Since Shy Guys figured strongly in the kidnapping attempt on Mario as a child, they still appear as bad guys in his dreams. Likewise, the Phanto scenes are a nightmarish, mixed-up reimagining of feeling helpless while Kamek's minions tried to abduct him and the feeling of only being secure when he was on Yoshi (symbolized in his dreams by the key). Also, Birdo is a bizarre representation of Yoshi, hence the different colours, egg attacks, and fireballs. In his baby mind, he couldn't be sure if Yoshi was good or bad, so he reinvents him as a key bad guy rather than a good guy. Birdo's gender confusion stems from Baby Mario having been aware that Yoshi was male, but also thinking that Yoshi must be a "mommy" since it took care of him.
Mario's dreams are of an earlier event that he remembers.
It's all true. When Mario and Peach were 2, they were reading a book and started to fidget over it. They tore out the last page, and the evil frog king Mamu caught them. Luigi, in a monkey suit, got shocked and told their babysitters,
Imajin, Lina, Mama, and Papa (from Doki Doki Panic) goes into the book. Mario and Peach saw all of the chapters inside (Chapters 1 through 7) and Mario remembered them. After they were rescued, Mario kept the memory but forgot some stuff (example: he remembered the subspace stuff but not that lamps were used). He later dreamed the events of Super Mario Bros. 2 and thus came that game.
Shy Guys appear in Mario's dreams, because...
He'd met them as a baby before.
...Subcon was consumed by a black hole gradually, before Subcon be recreated into Sprixie Kingdom(like the supernova event in Super Mario Galaxy), Imajin Family called sleeping Mario, Luigi, Peach and Toad for helping evacuate most of residents as they as can possible. Yet the price is "WHOLE IMAJIN FAMILY" are vanished by the Luma recreation... Before the last moment the family only things can do is: let their adventures memory replace into on Mario & friends to fitting in history. Peach is befriended with Lina, to memorized her, Peach decided using picking & throwing radish in her Smash Bros attack.
