Script for selecting layers with the same name

Try this:

#target photoshop
var doc = activeDocument;
var curLayer = doc.activeLayer;
var layName = curLayer.name;
var selcList = new Array();

var layerSets = 0
try{doc.backgroundLayer}
catch(e){layerSets=1}

var fullList = getLayerSetsData ();

for(var i=0;i<fullList.length;i++){
    if(fullList[i].name.toLowerCase() == layName.toLowerCase()){
        selcList.push(fullList[i].id)
        }
    }

if(selcList.length>1){multiSelectByIDs (selcList)}
else{alert('There are no duplicate layer names')};

function doesIdExists( id ){// function to check if the id exists
   var res = true;
   var ref = new ActionReference();
   ref.putIdentifier(charIDToTypeID('Lyr '), id);
    try{var desc = executeActionGet(ref)}catch(err){res = false};
    return res;
}

function multiSelectByIDs(ids) {
  if( ids.constructor != Array ) ids = [ ids ];
    var layers = new Array();
    var id54 = charIDToTypeID( "slct" );
    var desc12 = new ActionDescriptor();
    var id55 = charIDToTypeID( "null" );
    var ref9 = new ActionReference();
    for (var i = 0; i < ids.length; i++) {
       if(doesIdExists(ids[i]) == true){// a check to see if the id stil exists
           layers[i] = charIDToTypeID( "Lyr " );
           ref9.putIdentifier(layers[i], ids[i]);
       }
    }
    desc12.putReference( id55, ref9 );
    var id58 = charIDToTypeID( "MkVs" );
    desc12.putBoolean( id58, false );
    executeAction( id54, desc12, DialogModes.NO );
}

function getLayerSetsData()
{
    //var count = 0;//set counter for multi-dimensional array
    var lyrSets = [];

    while (true)
    {
        ref = new ActionReference();
        ref.putIndex(charIDToTypeID('Lyr '), layerSets);
        try
            {var d1 = executeActionGet(ref)}
        catch (err){
            break;
            };

        var c2t = function (s){return app.charIDToTypeID(s);};
        var s2t = function (s){return app.stringIDToTypeID(s);};
        var lyrSet = {};

        lyrSet.type = d1.getInteger(s2t("layerKind"));
        lyrSet.name = d1.getString(c2t("Nm  "));
        lyrSet.id = d1.getInteger(s2t("layerID"));
        
        lyrSets.push(lyrSet);
        layerSets++;
    }; 
    return lyrSets;
};

Okay, that's what I thought would make more sense than selecting any layers that have the same name, like all the A's and B's in your example. It would be easier for just one set at a time.

Since it looks like your file has several different layer names that are duplicates, how to you want to select just one set of them - by using the current selected layer?