Chess kings move only one square, but they can move in any direction.
The sudoku Anti-King constraint is global, applying to every digit, and means each digit does not repeat in any cell a chess king’s move away.
In practice, standard sudoku rules already prevent digits repeating in a row or column, so the Anti-King constraint only eliminates diagonally-adjacent repeats. Standard sudoku rules already prevent digits repeating in a box, so the Anti-King constraint only eliminate anything for digits along box borders.
I think the world would likely be a slightly better place with no kings anywhere at all. I’m pretty republican in that anti-monarchy sense. But I’m not calling for violence! I just don’t believe any sort of unitary authority can safely exist, and would like all those who attempt such things to be stopped from doing so.
Around this grid, we’ve embraced revolution, rejecting monarchy. No commonwealth, no unitary authority at all. So no digit repeats within a chess King’s move. Of course, normal Sudoku rules already eliminate repeated digits in orthogonally-adjacent cells, so really an Anti-King constraint only eliminates repeated digits in diagonal neighbors. Wacky kings!