{x, y, inv[x], inv[y], x ** y, 1, x ** inv[y], x ** x, y ** inv[x],
 y ** x, y ** y, inv[x] ** inv[y], inv[x] ** y, inv[x] ** inv[x],
 inv[y] ** x, inv[y] ** inv[x], inv[y] ** inv[y]}

{R1[x_,y_]:=x,R2[x_,y_]:=y,R3[x_,y_]:=inv[y],R4[x_,y_]=x ** y,...}

makeAssignment[wrapper_][e_, {i_}] :=
  (  ToExpression["R"~~ToString[i], InputForm, Hold]
  /. Hold[s_] :> Hold[s[x_, y_], e]
  /. Hold[l_, r_] :> wrapper[SetDelayed[l, r]]
  )

makeAssignment[Hold][x**y, {5}]

(* Hold[R5[x_, y_] := x ** y] *)

makeAssignment[Defer][x**y, {5}]

(* R5[x_, y_] := x ** y *)

makeAssignment[HoldForm][x**y, {5}]

(* R5[x_, y_] := x ** y *)

Head[%]

(* HoldForm *)

makeAssignment[Identity][x**y, {5}]

(* Null *)

??R5

(* R5[x_,y_] := x**y *)

$vs = { x, y, inv[x], inv[y], x ** y, 1, x ** inv[y], x ** x
      , y ** inv[x], y ** x, y ** y, inv[x] ** inv[y], inv[x] ** y
      , inv[x] ** inv[x], inv[y] ** x, inv[y] ** inv[x], inv[y] ** inv[y]
      };

$vs // MapIndexed[makeAssignment[Defer]]

(* {R1[x_, y_] := x, R2[x_, y_] := y, R3[x_, y_] := inv[x], ... } *)

$vs // MapIndexed[makeAssignment[Hold]]

(* {Hold[R1[x_, y_] := x], Hold[R2[x_, y_] := y], Hold[R3[x_, y_] := inv[x]], ... } *)

$vs // MapIndexed[makeAssignment[Hold]] // Apply[Hold] // Flatten

(* Hold[R1[x_, y_] := x, R2[x_, y_] := y, R3[x_, y_] := inv[x], ... ] *)

makeAssignment[wrapper_, e_, i_] := ...

makeAssignment[wrapper_][e_, {i_}] := ...

... // MapIndexed[makeAssignment[Hold]]

... // MapIndexed[makeAssignment[Hold, #, #2[[1]]]&, #]&

XYZ10 = 666;

Symbol["XYZ10"]
(* 666 *)

ToExpression["XYZ10", InputForm, Hold]
(* Hold[XYZ10] *)

Hold[V1, x + y] /. Hold[s_, e_] :> Hold[s[x_, y_] := e]
(* Hold[V1[x$_, y$_] := x + y] *)

Hold[V1, x + y] /. Hold[s_, e_] :> Hold[s[x_, y_], e] /. Hold[l_, r_] :> Hold[l := r]
(* Hold[V1[x_, y_] := x + y] *)

rhs = {x, y, inv[x], inv[y], x ** y}
lhs = {R1, R2, R3, R4, R5}
fun = MapThread[(Hold[#1[x_, y_] := #2]) &, {lhs, rhs}]

ReleaseHold /@ fun

fun = MapThread[(#1[x_, y_] := #2) &, {lhs, rhs}]