In southeastern England, northern France, and the Netherlands, from the twelfth to the fifteenth century, pretty much all lumber came from the local deciduous forest stock. Even for the cheapest, most disposable project. So your sample doesn't really tell us much.
[This phenomenon is actually still at play in 2012: something as disposable as pallets, of all things, from California, made out of redwood, of all woods, would be a total nonsense up here -- it's a rare and pricey wood up here, if you can manage to get some at all. But for them, who are probably drowning in redwood leftovers, it makes sense.]
So, back then, if you happened to live next to a forest of Brazilian Rosewood (one of the priciest woods out there) and that's what you had handy, you'd probably be building everything out of it, including pallets.
Another argument is that chests aren't exactly he best example to compare a cage with: they are generally designed to be long-lasting (which I admit the MitD's cage might well be), passed down the family, and if they can be esthetic as well as utilitarian, the better.
I have to admit I haven't read SoD yet... (shame on me...)
...so, the next step in that line of investigation, if we really want to push it that far ;), would be to try to determine what kind of forests would be growing in the place where that circus is. Do we have any clues as to the climate of the place in the panels shown? If we see trees, and it's all deciduous, then yes the crate is likely of a denser wood type, out of pure convenience. In Cliffport, for example, the wood stock from what we've seen is likely to resemble what was available in England, Holland, or northern France. A disposable crate in Cliffport isn't so unlikely to be oak.