Then how do so many mods include shape-changing, and why do so many people recommend clothing objects?
Shape changing from one shape to a completely different shape is always done by script. I think what da mage posted is the way how it's done. IIRC in Midas Magic the white tiger is a ridable creature. The player sits on top, but is invisible. You only see the tiger, even though the player is still there. And you can move the tiger without any additional scripts. Attack damage is done with a triggerzone in front of the tiger's mouth.
Clothing objects can be used, but only if the shape fits the skeleton. A Frost Atronach for example is possible, since the shape is roughly that of a human (there is a mod that adds a bodysuit like that on Nexus). A lich, a Xivilai, all possible. A mountain lion is not possible unless you make it stand on two legs, have arms and the like. Basically what a Khajiit is.