Consider two $m \times n$ matrices $A$ and $B$, where $m > n$. The Singular Value decomposition of $A$ and $B$ can be given as:
$A = U_A\Sigma_AV_A^T$ and $B = U_B\Sigma_BV_B^T$
The left and right singular vectors of $A$ and $B$ are same. i.e. $U_A =U_B$ and $V_A = V_B$.
Given the above conditions, is it possible to quantify the distance $A$ and $B$.
Also in general are there measures that quantify distance between any two $m \times n$ matrices ?
As mentioned in the related thread, the choice of metric should be made with application in mind. One can always measure distance between two matrices of the same size, e.g., by folding them into vectors (stacking columns one upon the other) and using some vector norm. Whether or not this distance is useful for the task is another matter.
In your situation the question reduces to quantifying the distance between two diagonal matrices. Extracting the diagonal entries, we get two vectors of length $n$. Three common ways of measuring distance between vectors are $$\sum_{i=1}^n |x_i-y_i| \tag{$\ell_1$ norm}$$ $$\sqrt{\sum_{i=1}^n |x_i-y_i|^2} \tag{$\ell_2$ norm}$$ $$\max_{1\le i\le n} |x_i-y_i| \tag{$\ell_\infty$ norm}$$
