get attribute object without using isolated scope angularjs custom directive

Not sure how you even get any console log output. It's not possible with the way you are defining your directive. You are using directive as an element, however its definition states it to be used as an attribute. Change it to this:

myApp.directive('myDirective', function() {
    return {
        restrict: 'E',
        link: function(scope, element, attr) {
            var atrib = attr.select;
            console.log(atrib);
        }
    }
});

Again, you need to declare resrict property as E "element". If you omit it (happens if you just provide a link function) it's A "attribute" be default.

If you want to see new value in your console after every option change in select, you can do it by the following way.

<div ng-controller="MyCtrl">
    <select ng-model="selectBox" ng-options="item for item in items"></select>
    <my-directive select="{{selectBox}}"></my-directive>
</div>

JS code:

myApp.directive('myDirective', function () {
    return {
        restrict:'E',
        scope: {
            select: '@select'
        },
        link: function (scope, element, attr) {
           scope.$watch('select', function(newValue){
               console.log(attr.select); //changed every time after a new select
           });
    }
   }
});

function MyCtrl($scope) {
    $scope.items = ['test1', 'test2', 'test3'];
    $scope.selectBox = $scope.items[0]; //default select's value
}

I've attached JSFiddle example for you.