What kind of transformation when the aspect is changed for a rotated shape?
Let's talk about linear geometric transformations in homogeneous coordinates.  In your question there are mainly two kind of geometric transformations involved: planar rotation and planar scaling, the computation-friendly analytical re-definitions of which are found hereUnified frameworks of elementary geometric transformations.

The 2D homogenous rotation matrix that rotates 2D objects clockwise by $\theta$ radians is as below:

$$R=\left[
\begin{array}{ccc}
 \cos \theta &amp; \sin \theta&amp; 0 \\
 -\sin\theta &amp; \cos\theta &amp; 0 \\
 0 &amp; 0 &amp; 1 \\
\end{array}
\right]$$

A 2D scaling due to unequal aspect ratio $\rho$ in this case is a stereohomology with $x$ axis $(0,1,0)^T$ as the stereohomology interface (a line of invariant points), and the normal direction $(0,1,0)^T$ as its stereohomology center per definition 3.9 in page 12 and line No. 5 in Table 1, page 9 of the article Unified frameworks of elementary geometric transformations. Then its scaling matrix can be obtained as below:

$$S=\color{gray}{\left[
\begin{array}{ccc}
 1 &amp; 0 &amp; 0 \\
 0 &amp; 1 &amp; 0 \\
 0 &amp; 0 &amp; 1 \\
\end{array}
\right]+\left(\rho-1\right)\dfrac{\left[
\begin{array}{c}
 0 \\
 1 \\
 0 \\
\end{array}
\right]\cdot \left[
\begin{array}{ccc}
 0 &amp; 1 &amp; 0 \\
\end{array}
\right]}{\left[
\begin{array}{ccc}
 0 &amp; 1 &amp; 0 \\
\end{array}
\right]\cdot \left[
\begin{array}{c}
 0 \\
 1 \\
 0 \\
\end{array}
\right]}=}\left[
\begin{array}{ccc}
 1 &amp; 0 &amp; 0 \\
 0 &amp; \rho  &amp; 0 \\
 0 &amp; 0 &amp; 1 \\
\end{array}
\right]$$

When aspect ratio normally is $\rho=1$, suppose under this circumstance the original object has the homogeneous coordinates of $n$ points in $3\times n$ matrix $X$, and the transformed object in $3\times n$ matrix $Y$, then the geometric transformation between them is the rotation you have mentioned:

$$Y=R\cdot X$$

When the aspect ratio abnormally is $\rho\ne 1$, both $X$ and the $Y$ after rotation are as a matter of fact scaled along the $y$ axis:

$$X'=S\cdot X$$
$$Y'=S\cdot Y$$

Now you want to find the geometric transformation $T$  which satisfies $Y'=T\cdot X'$. Since:

$$Y'=S\cdot Y=S\cdot R\cdot X =S\cdot R\cdot S^{-1}\cdot X'$$

then

$$T=S\cdot R\cdot S^{-1}$$ 
$$=\color{gray}{\left[
\begin{array}{ccc}
 1 &amp; 0 &amp; 0 \\
 0 &amp; \rho  &amp; 0 \\
 0 &amp; 0 &amp; 1 \\
\end{array}
\right]
\cdot
\left[
\begin{array}{ccc}
 \cos \theta &amp; \sin \theta&amp; 0 \\
 -\sin\theta &amp; \cos\theta &amp; 0 \\
 0 &amp; 0 &amp; 1 \\
\end{array}
\right]\cdot
\left[
\begin{array}{ccc}
 1 &amp; 0 &amp; 0 \\
 0 &amp; \dfrac{1}\rho  &amp; 0 \\
 0 &amp; 0 &amp; 1 \\
\end{array}
\right]=}\left[
\begin{array}{ccc}
 \cos \theta &amp; \dfrac{\sin \theta}{\rho } &amp; 0 \\
 -\rho  \sin\theta &amp; \cos \theta&amp; 0 \\
 0 &amp; 0 &amp; 1 \\
\end{array}
\right]$$

It is a rotation under the generalized meaning.