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)