One could view that in terms of unavoidable sets. The uniqueness rule says that if a placement of a digit in a cell would leave an unavoidable set, then that placement is impossible.
For example, if the possibilities for four cells are
{1,2} . . {1,2}
...........
..........
{1,2} . . {1,2,3}
then the 3 must go in the {1,2,3} cell, because if it doesn't then we are left with an unavoidable set and two solutions.
If you phrase the uniqueness rule in this way, it holds in any variant. It's just that the unavoidable sets for ordinary sudoku may not be unavoidable sets in the variant. For example, suppose we have a sudoku X grid. An unavoidable set in this grid for ordinary sudoku that does not intersect the diagonal or the back-diagonal will also be an unavoidable set for sudoku X. But an ordinary sudoku unavoidable that intersects the diagonal in one place will not be an unavoidable for sudoku X.
