Summary
3D games are not fundamentally different, but are less common because they are usually harder for people to understand the basic game mechanics. The main problem with 3D chess is that spaces on the board are adjacent without easy visual indication of such.
Full Explanation
Let's start with games with discrete spaces (like Chess, Risk, etc), the important principle is not the "dimensionality" of the board (2D vs 3D), it's the graph of the spaces on the board. 2D games usually have a planar graph, whereas 3D games are usually made that way to intentionally have a non-planar graph. They could still be made on 2D boards, but are not for the sake of novelty. There are still plenty of examples of games with 2D boards that still have non-planar graphs, such as the following:
The thing about non-planar graphs is that they are harder to reason about because there the visual cues for adjacency of spaces are either weaker or abstract. Edge detection is a key facet of both the neuroscientific study of human vision and the machine learning approach to computer vision. Thus, having adjacent territories share an edge means that you can have a human "understand" that adjacency in the Occipital Lobe. Two adjacent territories that are physically separate and only indicated to be adjacent on the board need to make it through the Occipital lobe and to the Cerebrum for someone to understand the adjacency, though there are at least high level visual cues they can use to help.
Two adjacent territories that are adjacent by means of game rules and not visually indicated at all or using something other than color (such as "vertical overlay plus 2D adjacency" in your 3D chess example) are probably the hardest to handle because there is no support whatsoever from the visual cortex, and this is purely effortful abstract reasoning. Effortful abstract reasoning is a major obstacle to gameplay when present in the basic mechanics. Take the classic Diplomacy map as an example of how much visual cues matter: many players miss the fact that Norway and St. Petersburg are adjacent because the border between them isn't very wide, even though the rules for adjacency are quite unambiguous.
Games that are popular tend to be ones that people get "intuitively". This is a really hard concept to define, but is best understood through examples. If, in Magic the Gathering, a goblin could fly and an angel couldn't, this would be really hard for people to remember (even though there is no rules reason this could not be the case), whereas the other way around is quite easy. If Settlers of Catan used species of plankton as the five resources, it would be an identical game, but would be far harder for people to remember what resource did what (since plankton have nothing to do with building roads and cities) and would consequently be massively harder to play. Similarly, that physically adjacent spaces allow movement between then is easy to understand, whereas physically disjoint spaces with no visual indicator allowing movement between them is hard.
Even in regular chess, the knight is usually the hardest piece for new players to "visualize" the movement of due to the fact that it cheats adjacency in it's movement, and consequently has fewer visual cues as to how it moves. Try playing chess on a board that is all one color (instead of alternating colors) and see how hard it is to visualize the movement of bishops.
The 3D chess game you give as the example is most likely unpopular primarily due to all of the reasons above making moving non-intuitive. The rules of the game are intentionally complex to take advantage of the novelty of the 3D structure of the board. While this is certainly entertaining for a game or two, it leads to a game that is more about not making mistakes than one with interesting strategic analysis, since you have to spend so much cognitive effort even understanding what moves are legal. From what I can find about the rules the proper way to visualize a move is to look at the board from above, but due to the overlap of the boards, this requires you to look at the board from a number of sides to see what is obscured by the bird's eye view. Add on top of this the fact that some of the boards can move to different positions and intuitive adjacency goes out the window. The lack of portability of the game is a minor inconvenience in comparison.
All of the above has been with regard to games with discrete spaces. There are plenty of examples of 3D games that are very successful that aren't about spaces:
The success here is that they don't require reasoning about complex 3D structures; they're just object manipulation and the human intuition about physics. Blacksmith puzzles (also known as tavern puzzles) are the counterpoint to this, but these are intended as solo exercises rather than having competitive rules draped on top of them.
