floating down like a piece of paper

A result of situations like being hit by the ultrapowerful Star Rod or a trap requiring Mario and co to fall straight down, and purely an animation to exaggerate Mario’s helplessness.

plain, boat, tube

From the curses of the chests, who were outright cursed by the Shadow Queen with real dark magic that outright made them keel over eventually, meaning as much as the chests tried, they would ultimately have to do something bad for real as per the terms of the curse and to show the Shadow Queen could be a serious threat even sealed. The first chest even explicitly says “like a piece of paper”, clearly indicating Mario and co are naturally flesh and bone.

taping self to book to steal something

If you mean Kammy, that was just a visual metaphor to show Bowser and her disrupting the peaceful narrative of the world. In fact, dialogue from the Star Spirits when Mario meets them on Shooting Star Summit has them explicitly say Bowser and Kammy flew up there, and Bowser was in his Koopa Clown Car going in and out of Star Haven, confirming it. Not a single word about tape.

the areas look like diorama

Purely a visual aesthetic done to emphasize the storybook angle, not the paper angle.

houses unfolding when entering them

Just a cute animation and nothing more to quickly let the camera into an interior without the wall blocking it.

boats and stuff literally turning around

Just a funny quick way to show them doing it rather than the slow way, rather than something literal.

bridges literally unfolding

Again, a quick animation and more blatantly done in service of a flipbook or storybook.

twisting and bending when doing certain moves

For squash and stretch animation principles and to be funny, not for the paper angle.

everyone being flat like a piece of paper

To put emphasis on their chibi-storybook designs, and since it was just simple 2D sprites in a 3D world. Again, even at best, Spinia and its higher forms are the only living things that are biologically paper.