In “Projection of one vector onto another”, the formula for magnitude used is

but actual formula is

Then in the video, it is wrong or there is any intuition behind this?

Yes, the given calculation is incorrect.

Please mark it on your side.

It should have been:

||y||^{2} = 2^2 + 2^2 + 1^2 = 9 (without the square root)

Or ||y||^{2} = (\sqrt{2^2 + 2^2 + 1^2})^2 = 9