Is there an objective measure of the superiority (or otherwise) that the strongest computer programs enjoy over the strongest human players which can be compared across the four main forms of chess, namely western chess, xiang qi (Chinese chess), shogi (Japanese chess), and makruk (Thai chess)? For example, can it be said that on comparable Elo scales the programs have say an x point advantage in western chess, a y point one in Chinese chess, and lag behind in the other two games by z points?
I would have to answer no in my opinion. The four forms of chess are different. Shogi from what I remember is a form of chess where capturing an opponent's piece allows you to, at a time of your choosing, put one of that piece on the board as yours. Now that is a level of tactical complexity that surely gives any engine a huge advantage over any strength of human. I have played a little Xiang Qi and I would say the same; in the 90's when Chinese players played Professional western chess for the first time they were awesome tacticians; Xiang Qi as their previous game being very tactical led to that.
While it is possible to write engines that can play different chess-family games, doing so generally yields an engine that are only "competent" at each one, at best. The best engines for each individual game incorporate specific knowledge and optimisations for that game, and are usually incapable of playing the others.
Many standard chess engines are not even capable of playing Chess960 correctly, because there are special rules for castling due to the variety of starting positions, even though the rest of the game is identical. However, adding this capability does not usually weaken the engine in the way that adding support for Shogi would.
Shogi, in particular, is very difficult to adapt a standard chess-playing engine to play well - even "fairy chess" engines which can support Xiangqi relatively easily. The immediate stumbling block is generally the piece-dropping mechanic, which completely explodes the branching factor of the game tree - so it is very difficult for an engine to explore the game tree to sufficient depth to treat the game in the tactical manner that engines usually do best at. Instead, Shogi has to be treated as a positional game, requiring a sophisticated positional analysis function to determine the correct lines to explore. Human players are naturally better at acquiring this knowledge than computers.
Conversely, a good Shogi engine would need to be taught about the special en passant and castling moves in standard chess, which don't exist in any form in Shogi, as well as the way pawns capture differently from moving, and the way in which pawns can promote to several different pieces rather than just one. All of the specialised Shogi positional knowledge that made it a good Shogi engine would then be useless for playing standard chess.
So the correct approach is to measure any given player's or engine's strength on an individual game. This will generally not be correlated strongly with its strength on a different game, if it is even capable of playing it.