Given list of associations, generate a list of all common associations

gather[x_] := {#[[1, 1]], #[[All, -1]]}& /@ Join[GatherBy[x, First], 
 GatherBy[{#[[All, 1]], #[[1, -1]]}&/@ Select[GatherBy[x, Last], Length@# >1&], First]] /.
 {{} -> Sequence[], {a_, {y_}} :> {a, y}}

gather @ list

{{AA, 3}, {AB, {2, 4}}, {BA, {2, 5}}, {BB, 6}, {{AB, BA}, 2}}

gather @ list2

{{{15, "->", 15}, {3, 8}}, {{16, "->", 16}, {2, 3, 4, 5, 6, 7, 8}}, {{{15, "->", 15}, {16, "->", 16}}, {3, 8}}}

Here is an alternative approach. It relies on Merge. It works with the example, but I'm not sure it's fully general.

It's convenient to start with a list of rules:

rlist = Rule @@@ list

Here are the functions used in the merges:

fun1 = If[Length[#] == 1, #[[1]], #] &
fun2 = Sort[#][[-1]] &

Here's the main part:

List @@@
 Normal @
  Merge[{
   Normal @ Merge[rlist, fun1],
   Reverse /@ Normal @ Merge[Reverse /@ rlist, fun1]
  },
  fun2
 ]

list = {{AA, 3}, {AB, 2}, {AB, 4}, {BA, 2}, {BA, 5}, {BB, 6}};
list2 = {{{15, "->", 15}, 3}, {{15, "->", 15}, 8}, {{16, "->", 16}, 
    2}, {{16, "->", 16}, 3}, {{16, "->", 16}, 4}, {{16, "->", 16}, 
    5}, {{16, "->", 16}, 6}, {{16, "->", 16}, 7}, {{16, "->", 16}, 8}};

list3 = {{{1, "->", 1}, 8}, {{3, "->", 3}, 8}, {{10, "->", 10}, 
8}, {{12, "->", 12}, 8}, {{13, "->", 13}, 2}, {{13, "->", 13}, 
7}, {{13, "->", 13}, 8}, {{14, "->", 14}, 6}, {{14, "->", 14}, 
8}, {{15, "->", 15}, 3}, {{15, "->", 15}, 8}, {{16, "->", 16}, 
2}, {{16, "->", 16}, 3}, {{16, "->", 16}, 4}, {{16, "->", 16}, 
5}, {{16, "->", 16}, 6}, {{16, "->", 16}, 7}, {{16, "->", 16}, 
8}};

  fun[lista_List] := Module[{as1, as4,as5}
  ,
  as1 = GroupBy[lista, First -> Last];
  as4 = Select[Reverse@GroupBy[lista, Last -> First], Length[#] > 1 &];
  as5 = Merge[Association[Thread[Subsets[Lookup[as4, #], {2}] -> #]] & /@ Keys@as4, Identity];

  {#[[1]], Sort[#[[2]]]} & /@  Transpose[{Keys[#], List @@ Normal @@@ #}] &@ Merge[{as1, as5}, Identity]]


fun[list2]

gives me :

{{15, "->", 15}, {3, 8}}, {{16, "->", 16}, {2, 3, 4, 5, 6, 7, 8}}, {{{15, "->", 15}, {16, "->", 16}}, {3,8}}}

while

Grid@fun[list3]

gives me:

{{{1, "->", 1}, {8}}, {{3, "->", 3}, {8}}, {{10, "->", 10}, {8}}, {{12, "->", 12}, {8}}, {{13, "->", 13}, {2, 7, 8}}, {{14, "->", 14}, {6, 8}}, {{15, "->", 15}, {3, 8}}, {{16, "->", 16}, {2, 3, 4, 5, 6, 7, 8}}, {{{15, "->", 15}, {16, "->", 16}}, {3, 8}}, {{{14, "->", 14}, {16, "->", 16}}, {6, 8}}, {{{13, "->", 13}, {16, "->", 16}}, {2, 7, 8}}, {{{1, "->", 1}, {3, "->", 3}}, {8}}, {{{1, "->", 1}, {10, "->", 10}}, {8}}, {{{1, "->", 1}, {12, "->", 12}}, {8}}, {{{1, "->", 1}, {13, "->", 13}}, {8}}, {{{1, "->", 1}, {14, "->", 14}}, {8}}, {{{1, "->", 1}, {15, "->", 15}}, {8}}, {{{1, "->", 1}, {16, "->", 16}}, {8}}, {{{3, "->", 3}, {10, "->", 10}}, {8}}, {{{3, "->", 3}, {12, "->", 12}}, {8}}, {{{3, "->", 3}, {13, "->", 13}}, {8}}, {{{3, "->", 3}, {14, "->", 14}}, {8}}, {{{3, "->", 3}, {15, "->", 15}}, {8}}, {{{3, "->", 3}, {16, "->", 16}}, {8}}, {{{10, "->", 10}, {12, "->", 12}}, {8}}, {{{10, "->", 10}, {13, "->", 13}}, {8}}, {{{10, "->", 10}, {14, "->", 14}}, {8}}, {{{10, "->", 10}, {15, "->", 15}}, {8}}, {{{10, "->", 10}, {16, "->", 16}}, {8}}, {{{12, "->", 12}, {13, "->", 13}}, {8}}, {{{12, "->", 12}, {14, "->", 14}}, {8}}, {{{12, "->", 12}, {15, "->", 15}}, {8}}, {{{12, "->", 12}, {16, "->", 16}}, {8}}, {{{13, "->", 13}, {14, "->", 14}}, {8}}, {{{13, "->", 13}, {15, "->", 15}}, {8}}, {{{14, "->", 14}, {15, "->", 15}}, {8}}}