The Mushroom Kingdom, Koopa Kingdom, and all of the other Kingdoms in the Super Mario Games exist within the Center of the earth which are only connectable by Ancient Tunnels that had been turned into Pipes that accidentally connected to the Main Pipelines of Major cities all across the world.

The Center of the earth is so huge it has Its own Atmosphere, Oceans, mountains, and Volcanos. A Large Portion of the Dinosaurs that Lived on the surface escaped to underneath the Planet before dying in Places called "Nesting grounds" They had slowly but surely evolved into Creatures such as Yoshi and The Prehistoric Fish evolved into fish such as Cheep Cheeps.

The Super Mario Bros Super Show is an In-Universe Show that changes the names of Many characters to avoid copyright so they changed things like Peach to Toadstool and Bowser to King Koopa.

Of course, Y/n was always there to save the Princess in times of trouble but she can Handle herself with only An Umbrella and A Frying Pan.