There are two cases where singular values are identical. The first case is for the deterministic scenerio where you have $B=AA^T$ is an identical matrix. I think if $B$ is all ones in the minor diagonal It also has identical singular values.$$\sigma_1=\sigma_2=\sigma_2,...,=\sigma_N$$
Basically from a given matrix; to get a matrix which has identical singular values, we need to manipulate the given matrix such that the minimum singular value is maximized. If we find a solution then
the general case will be satisfied with equality. Such a matrix can be as I said the identitiy matrix which has full rank. However it is not the only solution. To maximize the minimum singular value gives another solutions too. The idea is that we need to remove any dependency between rows and colums of this matrix! This implies that a fully random matrix will also satisfy
It is the same thing with taking the fourier transform of a dirac delta function and the result in the frequency domain is flat. Similarly when you take the fourier transform of the white noise which is random then you also get a flat spektrum.
As singular values can be seen as the frequency domain from another point of view. In this domain we have indentity matrix $\rightarrow$ dirac delta function and random matrix $\rightarrow$ white noise