What are the idiosyncrasies which are worked around in these two MSE answers?

Question

In the question here, Mr.Wizard gives the following code:

ClearAll[withTaggedMsg]
SetAttributes[withTaggedMsg, HoldAll]

withTaggedMsg[exp_, label_: "When"] := Function[,
   Internal`InheritedBlock[{MessagePacket},
     Unprotect @ MessagePacket;
     mp : MessagePacket[__, _BoxData] /; !TrueQ[$tagMsg] :=
           Block[{$tagMsg = True},
         Style[Row[{label, HoldForm[exp], "=", exp, " "}, "  "], "SBO"] /. tag_ :>
           MapAt[RowBox[{ToBoxes @ tag, #}] &, mp, {-1, 1}] // Identity
       ];
     #
   ],
  HoldAll]

as a way to add more information to messages generated during a long output. Here F'x extends this answer to generate the stack at the point of message creation:

SetAttributes[withTaggedMsg, HoldAll]
withTaggedMsg[] := 
 Function[, 
  Internal`InheritedBlock[{MessagePacket}, Unprotect[MessagePacket];
   MessagePacket[name__, BoxData[obj_, form_]] /; ! TrueQ[$tagMsg] := 
            Block[{$tagMsg = True}, 
     Identity@
      MessagePacket[name, 
       BoxData[RowBox[{ToBoxes@
           Style[Row[{"With stack", Stack[][[3 ;;]], Spacer[5]}, " "],
             Black], obj}], form]]];
   #], HoldAll]

My question is, can you please deconstruct what is going on here (subexpression by subexpression)? Please also pay extra attention to what Mathematica idiosyncrasies are worked around in those answers? And more specifically,

• Why the identity map?
• Why the need for a self referential rule? Thanks!