Basically unrelated. He's using geometry shaders to generate voxel mesh details-- perhaps with some LOD optimizations, while Nanite is a GPU-driven rendering technology that adapts triangle density to aim for a given fidelity target.
Nanite can use displacement maps and perform tessellation, but it uses an alternate pathway that's not necessarily more efficient than feeding it a high-poly asset to render.
I'd be very surprised if they were actually using geometry shaders (at least this conclusion can't be drawn from their post). Geom shaders are basically dead weight, a holdover from a bygone time, and are better avoided for modern renderers.
The techniques they're drawing on mentioned in the post - parallax mapping, shell mapping - do not generate explicit geometry, rather they rely on raymarching through heightfields in the frag shader. It's more likely that they're doing something like that.
Err, right, they're doing the transformations CPU-side. The blog hints that it's related to shell maps, so maybe the new mesh geometry is densest on the sharp edges?
Parallax mapping breaks on the edges of polygons though, while this technique seems to actually add geometry since the edges of surfaces are appropriately detailed and bumpy.
Nanite can use displacement maps and perform tessellation, but it uses an alternate pathway that's not necessarily more efficient than feeding it a high-poly asset to render.