It appears that the eigenvalue distribution is an attractor. That is, for a broad range of different input models (distributions of the random matrices), you get the same output--the same eigenvalue distribution--as the sample size becomes large. This is interesting, and it's hard to prove. (At least, it seemed hard to prove the last time I looked at it, about 20 years ago, and I'm sure that it's even harder to make advances in the field today!)
Now, to return to the news article. If the eigenvalue distribution is an attractor, this means that a lot of physical and social phenomena which can be modeled by eigenvalues (including, apparently, quantum energy levels and some properties of statistical tests) might have a common structure. Just as, at a similar level, we see the normal distribution and related functions in all sorts of unusual places.
Consider this quote from Buchanan's article:Recently, for example, physicist Ferdinand Kuemmeth and colleagues at Harvard University used it to predict the energy levels of electrons in the gold nanoparticles they had constructed. Traditional theories suggest that such energy levels should be influenced by a bewildering range of factors, including the precise shape and size of the nanoparticle and the relative position of the atoms, which is considered to be more or less random. Nevertheless, Kuemmeth's team found that random matrix theory described the measured levels very accurately.
That's what an attractor is all about: different inputs, same output.
Andrew Gelman on random matrices, the rest here