Earlier quoted context omitted.
The "dimensions" in these board games isn't a mathematical/topology thing, is it? Normally one dimension = one real number space. Every board game ever would fit in 1D then, "2D" chess included. I'm fine calling Backgammon 1.5-D. Physically you focus on a single dimension, and the second one matters too but it's not the same.
That's a good point, you could surely model full chess in a single dimension, it would just be that each pieces' movement rules would be more confusing E.g. a pawn can move exactly 8 squares towards its opponents end (16 on its first move if no piece occupies 8 squares away), but can only capture 7 or 9 squares forward (with some extra modulo math to prevent wrapping)
Yeah and it'd be even worse if you want to flatten out the piece colors and types into the 1D array.