How to add a list of vectors (lists) to a directed graph as an edge weight?

I think this is worth reporting to support. A workaround is to use a wrapper to prevent EdgeWeight from interpreting a list as an edge specification:

Block[{Identity},
    Graph[
        {1\[DirectedEdge]1, 1\[DirectedEdge]1, 1\[DirectedEdge]1},
        EdgeWeight -> Identity /@ {{1,0},{0,1},{1,1}}
    ]
]

If I want to display the EdgeWeights, there are also issues.

Using Carl Woll's answer to generate a graph with edge weights and using a variant of the accepted answer in the linked q/a:

ClearAll[displayWeightedMultiGraph]
displayWeightedMultiGraph = Module[{i = 1, j, g = #, bcurves,
  labels = PropertyValue[#, EdgeWeight], 
  gccoords = Cases[ToBoxes[#], GraphicsComplexBox[x_, y_, z___] :> x, Infinity][[1]]}, 
  bcurves = Cases[ToBoxes[g], {dir___, ar : Longest[__ArrowBox], ___} :> 
    (## & @@ Thread[{dir, {ar}}]), Infinity] /. 
  {ArrowBox[BezierCurveBox[x_, y___], z___] :> 
     Arrow[BezierCurve[x /. k_Integer :> gccoords[[k]], y], z], 
   ArrowBox[x : {__}, y_] :> Arrow[gccoords[[x]], y]}; 
   SetProperty[g, EdgeShapeFunction -> ({j = i++; Text[labels[[j]], 
     BezierFunction[#, SplineDegree -> 7][0.5]], bcurves[[j]]} &)]] &;

Examples:

g1 = Block[{Identity}, Graph[{1 -> 1, 1 -> 1, 1 -> 1}, 
      EdgeWeight -> Identity /@ {{1, 0}, {0, 1}, {1, 1} }]];
displayWeightedMultiGraph @ g1

g2 = Block[{Identity}, Graph[{1 <-> 2, 1 -> 2, 1 -> 2}, 
      EdgeWeight -> Identity /@ {{0, 0}, {0, 1}, {1, 0}}]];
displayWeightedMultiGraph @ g2