Good observation. Considering stacking of pieces maybe 1.5D though.

Chess has different pieces, which has higher entropy than a true 1d backgammon or 1d checkers with only one piece a field.

You could play with pieces that have a value of 1..N instead. Starting with 2,3, and 5 value pieces, and splitting them as needed. Making it one-dimensional again, while keeping 100% of the rules.

Final verdict, therefore: backgammon is 1D, not 1.5.

We could pretend that the second dimension was not playing a role in tactics back then, since it was very recently invented, like the brothers Wright invented the third dimension a hundred years ago. Or some hot air balloon at a world faire did it.