Obviously you missed a crossed circle and every tesselation of regular polygon has at least one of those. And that's just a start. Metric spaces are a huge area of mathematics.
1) Question did not mention distinct points. Two joined equal length segments with overlapping points qualify.
2) Question did not mention metric used, you assumed euclidean distance.
3) There are many more solutions in multidimensional spaces even just for euclidean metric. Question did not specify 2D euclidean (but explanation sort of assumed that) nor even planar. Have fun finding these in hyperbolic geometry - current model of reality due to General Relativity. Or in quantized space.
4) Do infinite distances count as distinct? Zero distances are more obvious, but what about zero limits with different signs?
Mainly etc. Never ask a general question about geometry. ;)