"In 2d, orthogonal moves can access all of the board whereas diagonal moves can't. In 3d, the same holds true only if you restrict diagonal moves to 2d."
Exactly. That (1,1,0) 'diagonal' steps are 'color bound', while (1,1,1) 'unicornal' steps are color alternating (but access only half the space) should already tell you that there is a huge qualitative difference between the two. That by combining these two move on your 'Bishop' allows it to reach the entire 'board' makes it very un-bishop-like.
What you say about Kings touches on the topology of the board. When you have a regular lattice is a matter of choice which cells you consider neighbors. The usual 2d chess board as 8-neighbor topology; all squares touching only in corners are considered neighbors. So a King can move diagonally, and there are Bishops. But you could have declared only the upper left and lower right corner to be connected to the square that shares the corner. Then you would have defined a hexagonal board, where each cell only has 6 neighbors. (And, unlike 8-neighbor topology, it turns out that all 6 neighbors are equivalent there. In 8-neighbor topology four of the neighbors only connect to two other neighbors, and four of them to four others. That is why Rooks and Bishops are different pieces. But on a hexagonal board the distinction between Rooks and Bishops blurs.)
You can do the same things in three dimensions. If you declare (1,1,1) moves as going to non-neighboring cells, the King should not have these moves, and Unicorns would not be elementary sliders but riders.
Incorrect. Unicorns could only access 1/4 of the board. You would need 4 unicorns starting in specific cubes to be able to cover the entire board. The best way to visualize this is that in any 2x2x2 cube sub-region of the 3d board, a unicorn would only have access to 2 of the 8 cubes. They are restricted quarterly due to the extra dimension. The color pattern holds for 2d bishops as their moves are in any one plane, but they alternate along 3d diagonals, so colors would not be able to determine what cubes the the unicorn could access. You would need to color the board with 4 colors to show this, and it may not even be able to be in a normal quad-alternation pattern, because the coloring would have to be based solely on the 3rd diagonals.
As for the kings, the topology of the movement is not any different. Unicorn-cubes are connected by a corner the same way diagonal squares are. There is no reason for these not to be considered neighbors. The definition of a king or queen move is "any direction" and no direction that is a straight line is excluded. I guess one could suggest a variant where all unicorn moves are excluded period, and every piece is restricted to a 2d plane per move, but this would seem awkward. In my opinion the moves a king could make should form a perfect enclosed cube around the king, the same way on a 2d board the possible moves form an enclosed square. It's the exact dimensional equivalent. Otherwise, a king could be right next to a queen and not be in check! How awkward would that be! Normal support mates wouldn't work either. So while on its own, one can consider the two types of diagonals to be different, for the game to be playable, and more consistent with other pieces, they should be considered the same. Of course in physics they have to be different as accelerating triagonally vs diagonally would result in different vectors/G-forces/orbits...etc, in just a pure mathematical game they could be treated the same.
"In 2d, orthogonal moves can access all of the board whereas diagonal moves can't. In 3d, the same holds true only if you restrict diagonal moves to 2d."
Exactly. That (1,1,0) 'diagonal' steps are 'color bound', while (1,1,1) 'unicornal' steps are color alternating (but access only a quarter of the space) should already tell you that there is a huge qualitative difference between the two. That by combining these two move on your 'Bishop' allows it to reach the entire 'board' makes it very un-bishop-like.
What you say about Kings touches on the topology of the board. When you have a regular lattice is a matter of choice which cells you consider neighbors. The usual 2d chess board as 8-neighbor topology; all squares touching only in corners are considered neighbors. So a King can move diagonally, and there are Bishops. But you could have declared only the upper left and lower right corner to be connected to the square that shares the corner. Then you would have defined a hexagonal board, where each cell only has 6 neighbors. (And, unlike 8-neighbor topology, it turns out that all 6 neighbors are equivalent there. In 8-neighbor topology four of the neighbors only connect to two other neighbors, and four of them to four others. That is why Rooks and Bishops are different pieces. But on a hexagonal board the distinction between Rooks and Bishops blurs.)
You can do the same things in three dimensions. If you declare (1,1,1) moves as going to non-neighboring cells, the King should not have these moves, and Unicorns would not be elementary sliders but riders.