That's a way

    In[147]:= Clear[fidellisTerm, fidellisOver, fidellisMin]
    fidellisTerm[x_Symbol, l1_List, y_Symbol, l2_List] := 
     Inner[Times, Overscript[x, #]& /@ l1, Subscript[y, #]& /@ l2, Plus] /; AtomQ[x] && AtomQ[y] && Length[l1] == Length[l2] && 
       VectorQ[l1, IntegerQ]
    fidellisOver[x_] := Last[Last[Extract[x, Position[x, _Overscript]]]] /; !FreeQ[x, Overscript]
    fidellisMin[x_] := Plus @@ First[GroupBy[SortBy[List @@ x, fidellisOver], fidellisOver]]
    fidellisMin[x_, y_] := Block[{ex = fidellisMin[x], ey = fidellisMin[y], i0},
      i0 = {fidellisOver[ex]} \[Intersection] {fidellisOver[ey]};
      If[i0 =!= {}, 
       Print["Both terms contain the minimal overscript ", i0], (* else *)
       fidellisMin[x + y]
       ]
      ]

    In[153]:= fidellisMin[fidellisTerm[\[Upsilon], {6}, \[Zeta], {{5}}], 
     fidellisTerm[\[Upsilon], {3, 9}, \[Zeta], {{9}, {15}}]]

    Out[153]= 
    \!\(\*OverscriptBox[\(\[Upsilon]\), \(3\)]\) 
    \!\(\*SubscriptBox[\(\[Zeta]\), \({9}\)]\)

    In[154]:= fidellisMin[fidellisTerm[\[Upsilon], {3}, \[Zeta], {{5}}], 
     fidellisTerm[\[Upsilon], {3, 9}, \[Zeta], {{9}, {15}}]]

    During evaluation of In[154]:= Both terms contain the minimal overscript {3}

    In[155]:= fidellisMin[
     fidellisTerm[\[Upsilon], {6, 2, 7}, \[Zeta], {{9}, {15}, {12}}], 
     fidellisTerm[\[Upsilon], {8, 7, 4}, \[Zeta], {{81}, {83}, {69}}]]

    Out[155]= 
    \!\(\*OverscriptBox[\(\[Upsilon]\), \(2\)]\) 
    \!\(\*SubscriptBox[\(\[Zeta]\), \({15}\)]\)

    In[156]:= fidellisMin[
     fidellisTerm[\[Upsilon], {6, 2}, \[Zeta], {{9}, {9, 15}}], 
     fidellisTerm[\[Upsilon], {3, 9, 
       8}, \[Zeta], {{9}, {9, 15}, {12, 81}}]]

    Out[156]= 
    \!\(\*OverscriptBox[\(\[Upsilon]\), \(2\)]\) 
    \!\(\*SubscriptBox[\(\[Zeta]\), \({9, 15}\)]\)

    In[157]:= fidellisMin[
     fidellisTerm[\[Upsilon], {6, 3, 5, 
       8}, \[Zeta], {{9}, {9, 15}, {9, 15}, {12, 81}}], 
     fidellisTerm[\[Upsilon], {4, 10, 7, 2, 
       9}, \[Zeta], {{9}, {9, 15}, {12}, {1}, {15}}]]

    Out[157]= 
    \!\(\*OverscriptBox[\(\[Upsilon]\), \(2\)]\) 
    \!\(\*SubscriptBox[\(\[Zeta]\), \({1}\)]\)

    In[158]:= fidellisMin[
     fidellisTerm[\[Upsilon], {6, 3, 5, 
       2}, \[Zeta], {{9}, {9, 15}, {9, 15}, {12, 81}}], 
     fidellisTerm[\[Upsilon], {4, 10, 7, 2, 
       9}, \[Zeta], {{9}, {9, 15}, {12}, {1}, {15}}]]

    During evaluation of In[158]:= Both terms contain the minimal overscript {2}

the result section looks awkward in the editor; for your convenience as a picture:

• the fact that your terms (fidellisTerm[]) are Plus connected has been used in fidellisMin[x_] as well as in fidellisMin[x_, y_]
• consider not to redefine built-in symbols because that can be confusing because all notebooks run by default in the same context in a Mathematica session