Answer
First: It is not a paradox: it is just wrong. The reasoning is wrong.
About $\pi = 2$ he says: "Well clearly we are approaching the diameter of the circle". That is a statement that he doesn't prove and which is false.
The same problem arises with the $\sqrt{2} = 2$ when he says: "Well clearly this geometric construction approaches the diagonal of the square". How does he know that?
All that this proves is that we have to be careful when we talk about finding limits from purely looking at pictures.
"Just because the sun sets in the west doesn't mean that it has to rise in the west as well.
Edit: There are plenty of example of proofs that seem right, but turn out to be wrong when we go over them in more detail. Take for example the proof that for complex numbers
$$
1 = \sqrt{1} = \sqrt{(-1)\cdot(-1)} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$$
Here again, the argument is invalid because the rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ doesn't hold for complex numbers.
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