added_date is a column you must pay careful attention to the quotes used. In MySQL columns are delimited with backticks, but this is only necessary if they conflict with any reserved keywords.
The other problem is using both double quotes to define the string and within the string. If you do that you must escape the inner ones, like this:
$query = $this->db->query("SELECT DATE_FORMAT(added_date, \"%M\") AS Month, SUM(total) FROM tbl_order GROUP BY Month");
If you have a syntax highlighting text editor, which is pretty much essential these days, you'd see how the string highlighting stops immediately on what's perceived as a closing quote in your original code. That's a warning you've got a syntax error.
The other fix is to use two different kinds of quotes. MySQL's default mode permits either double or single quotes for string values:
$query = $this->db->query('SELECT DATE_FORMAT(added_date, "%M") AS Month, SUM(total) FROM tbl_order GROUP BY Month');
In practice it's usually better to use single quoted strings for queries for the singular reason that accidental SQL injection is prevented:
$-type variables are not interpolated in single quote mode.