Here are two theorems from my textbook: https://i.stack.imgur.com/nl5JY.jpg
Why would the projection of y onto W not just be y times the identity matrix? Since an orthogonal matrix times its transpose is the identity matrix.
I figure that if AB = I, then BA = I as well. This would imply that U(transpose)U = UU(transpose), since U(transpose)U = I.
The matrix $U$ isn't a square matrix (it's $n \times p$), so $U^T U$ and $UU^T$ don't even have the same size. So although $U^T U=I$, you can't conclude that $UU^T=I$ (unless $n=p$ of course, but this case is rather trivial when you're talking about projection onto a $p$-dimensional subspace of $\mathbb{R}^n$).