To know what class of fractal a mandelbox is, we need to know the basic classes of 3d fractals. Since I can't find such a classification online, I have built my own classification based on the symmetric fractals of a 3x3x3 cube (see here). These give 81 types that can be reduced to a table of seven class names:
The names 'tree' and 'sponge' should sound fairly standard, and 'cluster' is a good name I think (e.g. a star cluster). 'foam' is fractal bubbles and is airtight.
A class that was new to me is what I called a 'shell', it is branching planes and so is a shell around a tree (its inverse is a tree), this is much like real shells which encase animals which are usually 'trees'. Even a snail shell encases a tree since a spiral is a tree with just one branch each recursion.
Now that we have a clear set of fractal classes, it is amazing how well this set allows you to classify the natural world:
Imagine trying to describe that scene without the classifications above... or worse still, out of basic euclidean geometry.
Here's the same table as above, rotated and showing real world versions of these structures.
(for higher def see here)
So finally we can ask, what class of fractal is the mandelbox? Well my hunch is that at its core it is a void cluster, however at a larger scale it seems to approximate several of the classes: