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I'm wondering how the number of singular values of a rectangular matrix $X$ could be determined.
For a square matrix, according to Number of Singular Values, I just take the rank of $X*X$. But this equation obviously only holds to square matrices and not rectangular ones. How can this be approached?
Furthermore, because of over- or underdetermination of the characteristic polynomial a rectangular matrix shouldn't have unique singular values, right?
You misread the answer (and you linked to a different one). It is $X^*X$, not $X^2$, and it works for matrices of any size.
If your $X$ is real, it is just $X^TX$.
Not sure what you mean by "unique".
