Two eyes are required for ordinary depth perception without workarounds, but depth perception is not required for 3D vision.

Speaking as someone who naturally has little to no depth perception (amblyopia affecting my right eye), I still see 3D just fine, catch flying objects (balls, frisbees, etc.) just fine, and am not disabled in any significant way except an inability to see into those magic eye puzzles.

Two eyes give you the ability to detect parallax without moving your head; those of us without binocular vision detect parallax by slight (imperceptible) movements of our head. There are many other useful visual artifacts of our 3D world: closer things are larger than farther things, closer things obscure farther things, and so on. Our brains just construct 3D worlds based on those data rather than the simpler, more reliable stationary parallax detection that you two-eyed folks use.

People also use depth of field / focal blur to aid depth perception. This is a major reason why the brain has issues with fake 3d using separate images, you want to focus on a part of the image, but things get blurry if you don't focus on the screen.

Motion parallax is the name for that effect. It is definitely a clue in perceiving 3 dimensions. Relative size is another clue: if you know roughly how big something is and it looks smaller, you can assume it's far away. Level of detail, interfering haze, etc all are other cues.

But the difference between what each eye sees is also an important factor in 3D vision. Movies just make it the only factor, which is probably why they can be disorienting. My guess would be that it's more important for things that are close than things that are far away; something 3 inches from your face appears to be in drastically different positions if you cover alternate eyes, but distant mountains look about the same.