Which pair of two distributions are more similar? Suppose I have two pairs of distributions: distributions A and B in Pair 1, distribution C and D in Pair 2. There are non-parametric tests to determine if there is evidence to say that the distributions in each pair are different. However, suppose that I found significant evidence that distributions A and B are different from each other, and distributions C and D are different from each other, are there tests I can use to determine which pair is "more" different?
Or is this even a valid question?

I think you might be looking for some sort of divergence metrics. KL divergence is a popular one, even though it is not symmetric, which might be desirable in your case. Other divergences can be found here, and some of them are symmetric. Also, you mentioned non-parametric tests: have you considered using the KS-statistic? A larger statistic should indicate that the two distributions are more different.
I am not sure if there is a way to "test" if this difference between the two differences is significant. It seems to me that whatever you do you will have only one instance of this difference, but it seems to be reasonable to draw some conclusion from it (as long as your distribution comes from a suitable sample size).
