Why is it "on the inside" and not "in the inside"?
The noun inside, as opposed to the preposition inside, appears to be composed of a noun side, pre-modified by the preposition in.
The choice of preposition is driven by the nominal side morpheme, with which we normally use the preposition on. Metaphorically, the inside and the outside are the two surfaces either side of a dividing line, rather than two contained areas.
Consider:
- On the side
- On the left side
- On the right side
- On the upside ...
- On the downside ...
- On the inside ...
- On the outside ...
If you wanted to express that a fly was crawling on the inside surface of the sphere of a football, you would say :
It is crawling on the inside of the ball.
But if you wanted to express that the fly was flying inside the empty sphere of the ball you would say :
It is flying inside the ball.
'Inside' already means 'within'. So to say 'in the inside' is duplicating meaning.
As Araucaria said, we usually speak of something being "on the side" of something.
Sometimes the prepositional phrase "on the inside" describes something that is literally on a surface. For example, consider a pipe: I can say "there's rust on the inside" to express that the inside surface of the pipe has rust on it. If I instead wanted to talk about the contents of the pipe, "inside" could be used by itself, without any additional preposition before it: "there's water inside the pipe", or the preposition in could be used: "there's water in the pipe".
The word on does have a broader sense and does not just refer to things that are directly touching a surface. But the basic sense of on the inside can often be seen as referring to something's position relative to the internal surface of some object, as suggested by the Merriam-Webster entry for the expression (cited in user067531's now-deleted answer):
1 : on the inner side, edge, or surface of something
- The number 22 car tried to pass the leader on the inside (of the track).
So it may be that "in the inside" isn't used because "in" and "inside" are shorter ways to express the idea of something being in the space enclosed by a boundary.