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I think it would not have a shape, or would rather be a zero dimensional point. For it to be any shape, it would have to have features, but you've already defined this as the fundamentally smallest 'thing' so it can't have any features smaller than itself. But you could also probably convince me that it's a sphere. I'm not sure if mathematicians consider a sphere of infinitesimal radius to still be a sphere or not, but treating it as infinitesimal kinda makes sense to me even if it's actually finitely small (the Planck length?)
A more interesting question to me is, assuming positions in space are discrete, which I'm not sure follows from saying there's a smallest possible object, how are those 'voxels' arranged? I don't think that's necessarily equivalent to asking what the shape of the smallest object would be. Pixels on a screen are in a rectangular grid, but the actual elements are circles in some types of screens.
There are a number of shapes besides cubes that can fill 3D space, but do the voxels even have to all be the same shape? Are we even looking for a 3D tiling, or could it be 4D in spacetime, or even higher dimension if it turns out the universe has more than 4 dimensions? Does it have to tile at all, or could it be entirely irregular while still being discrete? Is there any conceivable experiment that could prove any of these things, or is it unknowable?
What you're talking about sounds similar to the Planck length to me. I'm not a string theorist, but my understanding is it is well defined in normal 4D spacetime (where Planck time would be the time it takes a photon to travel one Planck length). Planck length is based only on universal constants (Planck's constant, speed of light, and the gravitational constant), and so any "thing" smaller than that is unphysical.
I think the interesting question is how do we get continuous experiences, measurements, and observations from a spacetime that is fundamentally quantized.
If it is a sphere then, the question that comes to mind (and may in turn inspire the first question) is, how would they fit together? If you cluster spheres together, you always end up with space between the spheres.
Our Planck length reality voxel isn't made up of physical matter; it's much too small. It's basically just quantum field fluctuations. It probably wouldn't interact with the Higgs field either so stacking them together would be impossible.