Tyler, you are too fast: didn't give me a chance to answer first! Of course, I agree with everything you say. I wrote the following before seeing your answer (except for the last paragraph).
Since I introduced this choice, let me explain. But first, echoing André, taking equivalence classes would be a wrong choice even if it gave a category, which it doesn't. In fact, the word “representation” in this context is a convenient lie: we are not doing representation theory here, and we must not think at all in terms of equivalence classes. For example, isomorphisms between “representations” control signs in equivariant cohomology theory.
One point is to obviate set theoretic nonsense. It has become unfashionable, perhaps, to pay attention to this, but of course the collection of all finite dimensional representations is not a set, and for many purposes, such as taking colimits as you say, one does want a set.
A mathematical point is that different universes give different categories of G-spectra, and that matters enormously: change of universe plays an essential role in equivariant stable homotopy theory. This could be dealt with in other ways, but use of universes is convenient.
Actually, how essential a universe is, depends on which choice of a category of $G$-spectra one has in mind. For all choices, it is very convenient to work with $G$-vector spaces with a fixed given $G$-inner product. For orthogonal $G$-spectra, the fact that the category $\mathcal I$ of such $G$-inner product spaces is essentially small (equivalent to a small category) allows us to use it without actually specifying a universe, although one does obtain a different $\mathcal I$ for each choice of a set of irreducible representations (the complete universe, allowing all, being the most important).
For $G$-spectra in the sense of Gaunce Lewis and myself, and therefore for the $S$-modules of EKMM (Elmendorf-Kriz-Mandell-May) use of a universe is truly essential: $G$-spectra are obtained from $G$-prespectra as colimits over inclusions of sub $G$-inner product spaces of a universe. Such colimits make no sense without use of some device to ensure smallness. In this line of development, use of a universe seems truly essential. The linear isometries $G$-operad $\mathcal L$ is central to the construction of the smash product (and to lots of work in equivariant infinite loop space theory), and $\mathcal L(j)$ is the $G$-space of linear isometries $U^j\to U$, where $U$ is the universe in which one is working. It would be ludicrous to try to make sense of that without working in a universe.
As a philosophical point, it is essential to be eclectic in this area and to allow use of different categories of $G$-spectra, such as orthogonal and Lewis-May or EKMM, since there are many things that one can readily prove with one and not the other. For a comparison of these two and discussion of change of universe, see for example Mandell-May, Equivariant orthogonal spectra and $S$-modules.
That source explains how, in orthogonal $G$-spectra, one can actually work with one fixed universe, even the trivial one, and obtain equivalent categories as Tyler says. Hill-Hopkins-Ravenel took that observation from Mandell-May and ran with it. To be honest, I sometimes regret we made that observation; as Tyler notes, it can be a source of confusion, and it can sometimes obscure the mathematics. Here again it is wise to be eclectic and think in terms of both physical change of universe and “phantom” change of universe in terms of that observation.
