Algorithm to minimize distance variance between 2D coordinates
So here is a rather simple approach to solve this problem. You know that there are $O(n^2)$ different distances that matter. You can characterize the solution of your problem by the smallest and the largest distance ($d_\min$,$d_\max$) between your matched points. This gives you a set of $O(n^4)$ candidates. 

So suppose you want to test, if there exists a matching where all distances are between $d_\min$ and $d_\max$. In other words, you want to solve the decision problem. This can be modeled as a perfect matching in a bipartite graph. Let $(u_1,\ldots,u_n)$ be the points in the first list and $(v_1,\ldots,v_n)$ be the points in the second list. Consider the bipartite graph that has an edge $(u_i,v_j)$ if and only if the distance between $u_i$ and $v_j$ is contained inside $[d_\min,d_\max]$. If this graph has a perfect matching, then there is an point-to-point assignment for the given distance constraints. Such a test can be done by the Hopcroft-Kahn algorithm, which needs $O(n^{2.5})$. A slightly faster solution can be achieved with fast matrix multiplication.

So you are left with trying out the candidates. This gives a $O(n^{6.5})$ algorithm. You can save a bit of running time by trying out all $d_\min$ values but then do a binary search for each $d_\max$ value. Now you have to solve only $O(n^2 \log n)$ decision problems (you also have to sort all distances, which is dominated by the overall running time) and hence the running time is only $O(n^{4.5}\log n)$.