ClearAll[AnomalousCancellationQ2]
AnomalousCancellationQ2[frac : {i_?Positive, j_?Positive}] :=
 Module[{samedigits, idig, jdig, ff, p, q, r, tmp},
  idig = IntegerDigits[i];
  jdig = IntegerDigits[j];
  samedigits = Intersection[idig, jdig];
  ff = i/j;
  If[samedigits != {},
   r = {};
   Do[
    p = Flatten[Position[idig, s]];
    q = Flatten[Position[jdig, s]];
    p = FromDigits[Delete[idig, #]] & /@ p;
    q = FromDigits[Delete[jdig, #]] & /@ q;
    tmp = Select[Tuples[{p, q}], #[[1]]/#[[2]] == ff &];
    If[Length[tmp] > 0,
     r = Join[r, Join[#, {i, j, s}] & /@ tmp];
     ];
    ,
    {s, samedigits}
    ];
   r
   ,
   {}
   ]
  ]
ijs = Select[Select[Range[1, 9999], IntegerDigits /* FreeQ[0]], IntegerDigits /* DuplicateFreeQ];
res = Reap[
    Do[
     Do[
      num = ijs[[i]];
      den = ijs[[j]];
      out = AnomalousCancellationQ2[{num, den}];
      If[Length[out] > 0,
       Sow[out]
       ]
      ,
      {i, 1, j - 1}
      ]
     ,
     {j, Length[ijs]}
     ]
    ][[2, 1]];

tmp = Catenate[res];

sel = Sort@Select[tmp, IntegerLength[#[[3]]] == IntegerLength[#[[4]]] == 2 &];
Length[sel]
t = Take[sel, UpTo[12]];
Column[Row[{#3, "/", #4, " = ", #1, "/", #2, " by removing ", #5}] & @@@ t]
SortBy[Tally[sel[[All, -1]]], First]

sel = Sort@Select[tmp, IntegerLength[#[[3]]] == IntegerLength[#[[4]]] == 3 &];
Length[sel]
t = Take[sel, UpTo[12]];
Column[Row[{#3, "/", #4, " = ", #1, "/", #2, " by removing ", #5}] & @@@ t]
SortBy[Tally[sel[[All, -1]]], First]

sel = Sort@Select[tmp, IntegerLength[#[[3]]] == IntegerLength[#[[4]]] == 4 &];
Length[sel]
t = Take[sel, UpTo[12]];
Column[Row[{#3, "/", #4, " = ", #1, "/", #2, " by removing ", #5}] & @@@ t]
SortBy[Tally[sel[[All, -1]]], First]