The relationship between hexagonal and cubic grids runs deeper, and in that sense hex bishops do correspond to unicorns. See e.g. https://www.redblobgames.com/grids/hexagons/#coordinates-cube

The article you linked to describes how hexes on a hexagonal grid could be described with three coordinates, because the six sides of a hexagon give it three axes. It's because of this that there is a choice when it comes to which two axes will be used to represent spaces for a particular hexagonal game, and different games have represented coordinates differently. Anyway, this article tries to make a correspondence between hexagonal grids and 3D grids, because each can be described with three coordinates. That's interesting as far as it goes, but the question remains whether there will be any correspondence between how pieces move on a 3D board and how they move on a hexagonal board. Since the article is simply about geometry and not about Chess variants, it does not cover that.

On a 3D cubic board, a Unicorn should move an equal distance in all three dimensions. So, if it starts out at (0,0,0), it could move to spaces with coordinates like (1,1,1), (2,2,2), (-1,1,-1), and so on. Looking at the diagram for axial coordinates on a hexagonal grid, the spaces that a Bishop could move to from (0,0,0) are all at a distance of 1 in two dimensions and a distance of 2 in the other dimension. These distances do not match how a Unicorn moves on a 3D board, and in fact, the grid does not contain a single hex with coordinates matching any space a Unicorn could move to from (0,0,0) on a 3D board.

I will also note that the spaces a hexagonal Bishop could reach from (0,0,0) do not match those that a 3D Bishop could reach. A 3D Bishop would be able to move to a space that is the same distance away in two dimensions but a distance of zero in the other dimension. Looking at the Axial diagram, these are on the orthogonal rows passing through (0,0,0). In other words, the way a Rook moves on a hexagonal board corresponds with the way a Bishop moves on a 3D board, at least in terms of how the coordinates change.

On a 3D board, a Knight move should adjust one coordinate by 1, one by 2, and leave the third the same. Looking at the coordinates on the Axial grid, there are no coordinates in that relation to (0,0,0). Every coordinate with no change in one coordinate has the other two at equal distances from (0,0,0). These all match the spaces a hexagonal Rook could move to.

So, in terms of Chess piece movements, I do not see much correspondence between hexagonal boards and 3D cubic boards. The changes in coordinates that Chess pieces are capable of will not be the same on each board. And even with three coordinates, the hexagonal board is missing many spaces from a 3D board and does not seem to support the sorts of coordinate changes that Rooks, Knights, and Unicorns are capable of in 3D Chess. While it supports the sort of coordinate changes Bishops are capable of, these belong to the hexagonal Rook, which can reach every space on the board.

Given all this, I do not see any way to support the idea that hexagonal Bishops correspond to 3D Unicorns. My takeaway from this is that piece movement should be defined in terms of geometric relations and not in terms of numeric coordinate changes. By defining it in terms of geometric relations, I can use the same definitions for Chess pieces on square, cubic, and hexagonal boards, and the pieces will move appropriately on each board. But if I try to define them in terms of coordinate changes, six-sided polygons (hexes) and six-sided polyhedrons (cubes) will not work the same way.