This is clearly a bug. In addition to the weird fence shape, the scaling behavior is also changed: fence widths change when the plot is resized.

A work-around is to post-process the output to straighten the fence lines and make fence widths independent of image size:

    ClearAll[repairFences]
    repairFences[w_] := Replace[#,  Line[x_] /; Length[Dimensions@x] ==  3 :>
      (Line[{Offset[{-w 72 /2 , 0}, #[[1,1]]], Offset[{w 72/2 , 0}, #[[1,1]]]}]& /@ 
        Partition[x, 2]), All] &

    llp = 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|>
           ] ;

    Row[Show[repairFences[.3] @ llp, ImageSize -> #]& /@ {200, 500}, Spacer[10]]

[![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 (fence length is independent of image size):

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

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

Make fences .3 user units long (fence length changes with image size):

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

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

You can still use `IntervalMarkersStyle` to style the fences:

    llp3 = 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 -> <|"FenceStyle" -> Red|>];

     llp3 /. Offset[{a_, _}, {b_, c_}] :> {b + .5 Sign[a], c}
[![enter image description here][4]][4]

**Update:** using the second approach to make a function:

    ClearAll[modifyFenceWidth]
    modifyFenceWidth[arg : h_[w_] | w_] := # /.  Switch[Head[arg], 
         Offset, Offset[{a_, _}, b_] :> Offset[{ w  Sign[a]/2 , 0}, b], 
         Real, Offset[{a_, _}, {b_, c_}] :> {b + w/2 Sign[a], c},
         Scaled, Offset[{a_, _}, {b_, c_}] :> 
           {b + w Abs[Subtract @@ PlotRange[#][[1]]]/2 Sign[a], c}] &


***Examples:***

you can specify the fence width in user units, printer's points or as fraction of the plot width:

    Row[Labeled[modifyFenceWidth[#] @ llp2, 
         Style["modifyFenceWidth @ " <> ToString[#], 18], Top] & /@ 
       {.4, Offset @ 20, Scaled @ .1}, Spacer[10]]

[![enter image description here][5]][5]


  [1]: https://i.sstatic.net/reFb5.png
  [2]: https://i.sstatic.net/lk8k5.png
  [3]: https://i.sstatic.net/KUKX0.png
  [4]: https://i.sstatic.net/hLcqT.png
  [5]: https://i.sstatic.net/IM1l6.png