If I take the unit interval $[0,1]$, and distribute $n$ red points and $m$ blue points on it uniformly at random, the points have an order on the line, and such an order can be permuted by a group of permutations. No permutation is particularly favoured statistically, so each permutation occurs uniformly at random.
If, however, I use $[0,1]^{2}$ instead, what does a "permutation" of the points mean? They can be ordered in the vertical and horizontal direction, but is there a natural analogue of a permutation of spatial points when in 2$d$, or higher?
Permutation has nothing to do with dimension.
Label each of your points with numbers from $1$ to $n$. Choose a permutation $\pi$ of $1$ to $n$. Then for each $i$, move point $i$ to the former position of point $\pi(i)$.
This can be done in any number of dimensions, or without any concept of dimension whatsoever.