chrome.storage.sync.get($div.attr('id'),function(items){
    var lastError = chrome.runtime.lastError;
    if (lastError) console.log($div.attr('id')+" does not exist.n", lastError);
    else chrome.storage.sync.set({$div.attr('id'):$div.html()}, function(){}); //I'm having the error from this line
});

var myobj = {}, key = $div.attr('id');
myobj[key] = $div.html();
chrome.storage.sync.get(key,function(items){
    var lastError = chrome.runtime.lastError;
    if (lastError) console.log(key+" does not exist.n", lastError);
    else chrome.storage.sync.set(myobj, function(){}); //The error is gone
});

{$div.attr('id'):$div.html()}

else {
  var obj = {}, key = $div.attr('id');
  obj[key] = $div.html();
  chrome.storage.sync.set(obj, function(){});
}