The linear operator $T\in \mathcal{\mathbb{R}^2}$ defined by $T(x,y)=(2y,x)$ has singular value decomposition (SVD) $$T(x,y) = 2\langle (x,y), (0,1)\rangle (1,0)+1\langle (x,y),(1,0)\rangle (0,1).$$
However I would like to consider a "natural" way to express this fact in terms of matrices. I.e., something of the form $A=U\Sigma V^*$, where $U,V$ are unitary and $\Sigma$ is diagonal with singular values of $T$ on the diagonal.
This suggests that I take $\Sigma=\left(\begin{array}{cc} 2 & 0\\ 0 & 1 \end{array}\right)$, $V=\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right),$ and $U=I_2$. The product gives $$U\Sigma V^*=\left(\begin{array}{cc} 0 & 1\\ 2 & 0 \end{array}\right).$$
Of all matrix representations of $T$, why is the right hand side the matrix of $T$ in the standard basis? There are two bases involved (the columns of $U$ and the columns of $V$), so I don't see why the matrix on the right hand side only takes one of these bases into account.
I don't understand your question too well so it might be that this doesn't provide an answer. But let my try to address what I believe confuses you.
What you are doing with the matrices $U,V$ is you change the basis of your vector space $\mathbb R^2$ on both sides. Let $A$ denote a two-dimensional vector space. Then you are considering a linear map $f:A\to A$ and you represent it by a matrix $T$ which we interpret as linear map $T:\mathbb R^2\to\mathbb R^2$. This representation as matrix involves a choice of basis for $A$ on the left and on the right.
So what you are actually doing with singular vector decomposition is that you take a basis $B$ on the left $A$ and another one, say $B'$ on the right $A$ and write down, how $T$ looks with respect to these bases. Check the wikipedia article on base change for further details.
For your specific example, $B=\{(1,0)^t,(0,1)^t\}$ and $B'=\{(0,1)^t,(1,0)^t\}$. It can happen that one of these bases is the standard basis. But you are always taking a basis on the left and one on the right because otherwise you wouldn't be able to speak about a matrix.