I am trying to think what would be the starting point and a basic program to "categorize" the multiverse.
Let us begin with the accessible category MOD(ZF) of all models of ZF and usual maps of models. Now, obviously, there is something there that makes it distinguished from other analogous cat of models of first-order theories (say GROUPS).
As Joel has reminded us above, there is a plethora of technologies to build models out of existing ones, either by enlargement (forcing), or by restriction (inner models).
My immediate reaction (from the categorical standpoint) is this:
can we characterize these operations STRUCTURALLY?
I mean: can we, without resorting to the usual set-theoretical details of such constructions, describe them (for instance the various types of forcing enlargements) in a purely diagrammatic way within MOD(ZF)? That would be, or so it seems to me, step one (this step may require to introduce additional maps, in which case it can guide us toward the proper set-up).
Asssuming that one can partially answer in the affermative, step two would be:
axiomatize MOD(ZF) as an accessible category PLUS those structural operations.
Step three would be this: unlike other MOD(T) categories, MOD(ZF) has this fascinating property: given a model M of ZF, ie an element of the cat,
one can onsider the MULTIVERSE of internal models of M (let us call it the chinese boxes principle, if there is no name already).
That seems to suggest, as the PO has in fact hinted at, that MOD(ZF) is in fact a higher cat.
The game is not over. There is a subtle interplay between those internal models and their externalization. Here I have no clue, but it looks like the proper line of thought would be to use the machinery of internal category theory to express this feature.
Step four (excuse me, I am talking free-wheeling) would be: now, given the full axiomatization of Mod(ZF), is there some way to prove that is "unique"? Or are there other models of this axiomatization which are not MOD(ZF)? Most likely there would be other (multi)universes which are not the intended one.
POST SCRIPTUM: I am no expert, so the set-theory adepts should correct me if in the loose sketch of the program above I introduced some unintended nonsense. In particular, I assumed a broad threefold partition of model-building tools, into enlargement, restriction (such as L and variants thereof) and finally internalization (ie starting from a model M create another one which happens to be a set in M). If there is something which does not fit the mentioned labels,add it (them) to the menu. For instance, There are (at least) two types of enlargement, one which keeps the tallness fixed, and another type which makes the model taller. A structural approach to model building inside the multiverse has to account for all those types, so a preliminary accurate taxonomy of basic model building shapes is in order.