Close but not quite. The term "universality" is different in phase transition theory from what it means in random matrix theory (which is what's at play here), but they've got some similarities too.

In phase transition theories, you've got two different states (like liquid water and water vapor), and when you vary some high-level parameters (like pressure and temperature) you can go from one of these states to the other. This means that the only "interesting" physics (in the sense of "distinct from the well-understood liquid/gas behavior" can only happen right as you get near that transition, so that liquids are seamlessly becoming gases which are seamlessly becoming liquids again. The "transition" allows a lot of behavior not seen elsewhere, in fact all of the behavior "in the middle" between the two regimes. Because you've got this considerable mixing of the two states, often "zooming in" is the same as, say, adjusting the proportion of liquid to gas -- so since it all happens in the same space, you look for these "scale invariant" theories that are the same upon zooming in. Those theories then can't depend on too much particulars, but just depend on various symmetries, so they become "universal". See http://en.wikipedia.org/wiki/Critical_exponent .

Random matrix theory is similar, but for a different reason. The issue is, if you have a very complicated system that you can represent as a huge matrix, often the eigenvalues of the matrix tell you something concrete and physical. The example given above was the eigenvalues of a Hamiltonian matrix in quantum mechanics, which gives you an "energy spectrum" (discrete energies that the system can be at, so that it can e.g. absorb a photon of energy B - A to transition from a state of energy A to one of energy B).

Why would a Hamiltonian be random? You have to imagine a big molecule with lots of parts, not entirely under the control of the experimentalist. Maybe you've got a carbon nanotube hanging over a trench that you've etched under it, but the etching has caused other atoms to be stuck to the tube in unpredictable ways, and maybe the "islands" on either side couple to the nanotube in complicated ways.

Wigner discovered that very often these random variations break the symmetry between certain levels, so that where you once had 3 states at the same energy, now it's like those "energy eigenvalues" have "repelled" each other. He realized that the right way to start to think of the problem involved taking a matrix and adding random elements to it; this led to a nice set of models where you take it to the extreme and just randomize the entire matrix and look at its eigenvalue "density" rather than the exact levels. Wigner in particular discovered that this density function tends to look like a semicircle.

The essential similarity between these two is, you get to some point where "there's nothing more to say". With random matrices, when you specify how you're building the matrix and what sorts of properties it has to have, then you get the eigenvalue spectrum, and there's no other details to fixate upon. Similarly when you have a phase transition, the "we're taking on all the states in between gas and liquid" status of what you're doing means that the particulars of those states cannot matter. So in both cases it becomes just, "what's the configuration, what symmetries does it have" that determines how macroscopic parameters (whether critical exponents or eigenvalue densities) ultimately behave.