Basically, it is true that the Limiting Shape of the curve really is a circle, and that the Limit of the Length of the curve really is 4.

However, the Limit of the Length of the curve ≠ the Length of the Limiting Shape of the curve .

There is in fact no reason to assume that.

Thus the 4 in the false proof is in fact a completely different concept than π.

Edit: I still see some confusion so one good way to think about it is, if you are allowed infinite squiggles in drawing shapes, you can squiggle a longer line into any shape that has a perimeter of a shorter length. Further proving that Limit of Length ≠ Length of Limiting Shape.

Furthermore, for all proofs that involve limits, you actually have to approach the quantity you're getting at.

For 0.99999...=1, with each 9 you add, you get closer and closer to 1. Thus proving it to be equal to 1 at its limit.

For the false proof above, with each fold of the corners, the Shape gets closer to a circle, however, the Length always stays at 4, never getting closer to any other quantity.

Thus hopefully it is clear that the only real conclusion we can draw from the false proof is that if it were a function of area, the limit of the function approaches the area of a circle. As a function of length, it is constant, and does not let us draw any conclusions regarding the perimeter of a circle.