You have undoubtedly played them. Games with floating land and platforms, "falling out of the world", high jumping, floating and slowly revolving objects, and wingless flying enemies.
Question: What natural properties of the world or technologies put in place could cause these kind of things to exist on a planet? The atmosphere and biology of the planet should be Terran like where possible.
This approach relies more on theoretical mathematics than conventional physics (although in theory should work within Newtonian mechanics at least).
Take the core concept of a platformer: a 2-D model of a 3-D world, and add a fourth dimension. You will then have a world with 4 spatial dimensions (Lets call them W, X, Y, and Z). The primary inhabitants of this world should only be able to perceive and traverse 3 dimensions (X, Y, and Z), and therefore exist on a single slice of the 4th dimension (W=0).
The 'floating' platforms actually extend into the W dimension and are anchored to structures outside of the W=0 slice. Likewise, the floating items are tethered to these structures by a weaker link; when they are picked up, this tether breaks. Large bosses likewise extend into the W dimension, and can use this to affect an appearance of flight, just like the other floating objects.
Infinitely long falls could be explained by a distortion in 3 dimensional space, whereby X=100 redirects you do a different slice of the W dimension, which loops back to X=10 (basically, think portals).
High jumping could be explained by reduced gravity.
You use a combination of magnetic fields, superconductive metal platforms, and low gravity.
Super conductive platforms (made of dirt encrusted superconductive metals) on a super cold planet could take advantage of the Meinssner Effect to create "floating platforms". Of course, you'd eventually hit the ground if you fell from one, so you don't have infinite falls (which are impossible since you always fall towards the source of gravitational pull).
Low gravity allows the growth of massive creatures (huge bosses), and flying enemies are just creatures that you design in your world. The lower gravity, however, also does allow for higher jumping.
Don't forget my favorite, superconductive flux pinning. Macroscopic objects can be locked in space as if invisible intangible beams connected them into a rigid structure. I've detailed this in several previous answers.
This structure descibed in this link could suit this architecture just fine.
Now what about wingless animals? In this one I discuss a swarm of masses; hard science writeups of space construction using superconductors go into how they can be controlled dynamically. An animal would need to contain such a dynamic element and be able to control it.
If you wanted to move in such a way, you'd strap plates to your torso and have a computer control. A neuro interface is a step from there, and then we can implant them in other animals for our amusement.
Some of the physics suggested by Larry Niven's The Integral Trees, might help. Basically, he proposes an enormous torus of gases surrounding a star, with a narrow region near its center which is dense enough to support life. Into this planet-less ring of atmosphere he floats objects; in his case giant trees, in your case, platforms.
I don't know how realistic it is, but Niven certainly sold it well.
I've seen platformers that were simply placed on buildings or in a mine. So you have a wall on one side and open space on the other. But no one ever moves towards or away from the wall (because the wall blocks them from moving inward and they'll fall if they move outward). They only move alongside the wall. Except perhaps to jump around a platform, but we don't see that. From our two dimensional view, it seems like they move through platforms.
Usually the wall is on the far side from us, but you could also consider that the screen is the edge of the wall and we are looking out from the wall. That would give the normal appearance of platforms just hanging in air.
Your world might need to be low gravity to allow for the impossibly long jumps common in platformers. That might also make it easier to have massive but weak bosses.
Heavy atmosphere and ballooning species.
let's say you have a planet with a dense $Co_2$ atmosphere then plant life could use vacuoles or multicellular sacs filled with a lifting gas. If they are mat forming then the mats would be moving platforms.
If there are free floating small critters than larger filter-feeding things with flotation bladders could chase them about.
Heavy atmosphere and biological balloons would solve most of your requirements.
High jumping would be achieved through buoyancy.
