jQuery: value.attr is not a function

The second parameter of the callback function passed to each() will contain the actual DOM element and not a jQuery wrapper object. You can call the getAttribute() method of the element:

$('#category_sorting_form_save').click(function() {
    var elements = $("#category_sorting_elements > div");
    $.each(elements, function(key, value) {
        console.info(key, ": ", value);
        console.info("cat_id: ", value.getAttribute('cat_id'));
    });
});

Or wrap the element in a jQuery object yourself:

$('#category_sorting_form_save').click(function() {
    var elements = $("#category_sorting_elements > div");
    $.each(elements, function(key, value) {
        console.info(key, ": ", value);
        console.info("cat_id: ", $(value).attr('cat_id'));
    });
});

Or simply use $(this):

$('#category_sorting_form_save').click(function() {
    var elements = $("#category_sorting_elements > div");
    $.each(elements, function() {
        console.info("cat_id: ", $(this).attr('cat_id'));
    });
});

You are dealing with the raw DOM element .. need to wrap it in a jquery object

console.info("cat_id: ",$(value).attr('cat_id'));

Contents of that jQuery object are plain DOM elements, which doesn't respond to jQuery methods (e.g. .attr). You need to wrap the value by $() to turn it into a jQuery object to use it.

    console.info("cat_id: ", $(value).attr('cat_id'));

or just use the DOM method directly

    console.info("cat_id: ", value.getAttribute('cat_id'));