Edit: At least it seems to run well under Wine. I suppose a Linux version isn't too strong of a priority.
I've tried a wide variety of voxel modeling tools, free and not free, and this one's by far the most intuitive, and prettiest, I've found - in that it's the only one I haven't given up on in intense annoyance after 10 minutes or so.
Might end up using this for an animation project - thanks to the OP!
 Edit: I did some more digging... Looks like they used Unity to create Monument Valley: https://unity3d.com/showcase/case-stories/monument-valley
What I call a voxel is much smaller (on the order of a screen pixel), and drawn as a point sprite or screen-facing quad with a single color. Texturing is achieved by having many voxels with different colors/materials.
Here are some videos of what we used to call voxels:
DOS game: https://www.youtube.com/watch?v=51E_G7NCXVM
Gameboy game: https://www.youtube.com/watch?v=3zVy-4CEFQU
Medical imaging: https://www.youtube.com/watch?v=GrLqFNhVL68
Note that they were not using voxels because they had a certain artistic style to them. They used voxels because with the hardware they had, this was the most realistic rendering they could achieve. Once GPUs came out with hardware-accelerated triangle rasterization, voxel engines died.
For more information about point-sampling, see this classic paper, "A Pixel Is Not A Little Square":
And for a more in-depth discussion of sampling, albeit in one dimension, watch Chris Montgomery's video "Digital Show and Tell":
Voxels are "volumetric cells" just like pixels are "picture cells". The size of them has absolutely nothing to do with it, and never has. Big voxels are still voxels. The data in the file is specified as volumetric samples... Maybe the issue is the texture you're talking about, but I haven't seen any textures in any of the galleries, other than the few examples that render mesh edges in addition to the cubes. Where are you seeing textures?
Six square polygons enclosing a space is surely just as 'voxel' as meshes generated by marching cubes for medical imaging. I've worked in medial imaging, written more than one volume renderer and used used many more than that, for both games and research. As far as I'm concerned, Magica Voxel is both truly voxel and truly awesome, IMO.
Alvy Ray Smith's paper is great, but how to handle sampling is a different subject completely from whether the scenes are volumetric, which they are. These voxels do have a certain artistic style to them, and the aesthetic choice to give them sharp edges was made intentionally and called out explicitly, i.e., "8-bit".
Look at the file format. It is most definitely voxel based data. That it is comprised of regular sample points on a 3d grid is a matter of fact.
It takes all of 30 seconds to download the app and run it in order to see first hand it is voxel based, less time than it takes to write comments.
Maybe you've conflated the concepts of underlying data with the chosen method of rendering. But, even if that's why, it's false to say that rendering voxel data with polygons isn't voxels. Voxel data has been rendered with polygons since the term was invented, whether it's iso-surface (marching cubes), splatting, shear warp or cubes. The only time voxel data is not rendered using polygons is when it's ray traced.