"she was the child of his desire and the Countess hated her"

The Countess here fulfills the role of the 'wicked stepmother', who is something of a stock character in fairy tales; 'Snow White', 'Hansel and Gretel' and 'Cinderella' are good examples.  There are various theories behind the existence of the character.  In the past, a significant number of women died in childbirth, making it highly likely that another woman would raise the child.  It has been suggested that a stepmother may resent the constant reminder of her predecessor - whom her husband would otherwise still be married to - and deliberately favours her own children more.  She is also shown to be jealous of the youth and beauty of a stepdaughter who is blossoming into womanhood, no doubt reminding her father of the way her natural mother once looked.