This is clearly a bug.

A work-around is to post-process the output to fix the fence lines:


    ClearAll[repairFences]
    repairFences = Replace[#,  Line[x_] /; Length[Dimensions@x] ==  3 :>
        (Line[Thread[{#[[;; 2, 2, 1]], #[[1, 1, 2]]}]] & /@ Partition[x, 2]), All] &;

    repairFences @ 
       ListLogPlot[Table[{i, Around[1 + Sin[i], 0.1 + 0.1 Cos[i]]}, {i, 0, 10,  1.0}],
           PlotRange -> {All, {0.01, 10}},
           IntervalMarkers -> "Fences",
           IntervalMarkersStyle -> <|"FenceWidth" -> .3|>
           ] 


[![enter image description here][1]][1]

An alternative work-around is to remove `IntervalMarkersStyle -> <|"FenceWidth" -> .3|>` and post-process to modify the fence lines to desired length:

    llp2 = ListLogPlot[Table[{i, Around[1 + Sin[i], 0.1 + 0.1 Cos[i]]}, {i, 0, 10, 1.0}],
         PlotRange -> {All, {0.01, 10}}, IntervalMarkers -> "Fences" ] 

Make fences 20 printer's points long:

    llp2 /. Offset[{a_, _}, b_] :> Offset[{10 Sign[a] , 0}, b]

[![enter image description here][2]][2]

Make fences .3 user units long:

    llp2 /. Offset[{a_, _}, {b_, c_}] :> {b + .15 Sign[a], c}

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/0X9ZC.png
  [2]: https://i.sstatic.net/lk8k5.png
  [3]: https://i.sstatic.net/KUKX0.png