Ring-LWE elliptic gaussian distribution
(I am one of the authors of the paper you're asking about.  The isomorphism $\varphi$ you wrote is the intended one.)

The key observation is that a Gaussian $D_r$ of parameter $r$ over $\mathbb{C}$ is "spherical," i.e., it is the sum of independent Gaussians (both of parameter $r$) for the real and imaginary components, and so is invariant under rotations of the complex plane.  Therefore, $x \cdot D_r$ equals $D_{|x| \cdot r}$ for any $x \in \mathbb{C}$.

The claim then follows by the fact that multiplication in $K$ corresponds to coordinate-wise multiplication in the real/complex components of $\sigma$.  That is, the distribution of $x \cdot D_{(r_1,\ldots,r_n)}$ for $x \in K$ is just $D_{(|\sigma_1(x)| \cdot r_1, \ldots, |\sigma_n(x)| \cdot r_n)}$.  (Note that the conjugate symmetry of the pairs of complex embeddings is preserved, as needed.)