Highlight elements in the list using pattern matching

rg = RelationGraph[UnsameQ @ ## && Length@Intersection[##] >= 2 &, list]

hl = VertexList @ EdgeList @ rg

{{a, b, c}, {b, c, d}, {c, a, m}, {c, d, n}}

list /. x : Alternatives @@ hl :> Style[x, Gray] 

list /. x : Alternatives @@ hl :> Highlighted[x, BaseStyle -> Red]

HighlightGraph[rg, hl]

You can also use ConnectedComponents and select components with more than 1 vertex:

ccs = Select[Length @ # >= 2 &] @ ConnectedComponents[rg]

 {{{a, b, c}, {b, c, d}, {c, a, m}, {c, d, n}}}

list /. x : Alternatives @@ # :> Highlighted[x, BaseStyle -> Red]& /@ ccs

ClearAll[formatList]
formatList[list_] := Module[{rules},
  rules =
    AssociationThread[
      list -> (If[Max[#] >= 2, Gray, Black] & /@ 
        Function[{element}, 
          Length@Intersection[element, #] & /@ 
           Complement[list, {element}]] /@ list)
    ];
  Style[#, rules[#]] & /@ list
]

formatList[list]