Is it permissible to continue an engagement with a girl who allows her fiancé to kiss and hug her and to disclose to him her secrets and her family's secrets, with her mother's knowledge of this and her approval of going out with him without her father's knowledge, and is the fiancé in this case considered a betrayer of his old friend who wished to marry her?

The engagement is a marriage proposal. Islamic law permits the suitor to look at his fiancée, but she remains a stranger to him until the marriage contract is concluded. Therefore, it is not permissible to be alone with her, go out with her, touch her, or exchange words of love. Thus, your relationship with your fiancée, as you described it, is not permissible according to Islamic law and must be severed until the marriage contract is concluded. You can conclude the contract and delay consummation. As for proposing to a girl who has not been proposed to by your friend, your action is permissible according to Islamic law and is not considered a betrayal of him. Allah is not pleased with the actions of the girl you described, but this does not necessarily mean that she is unchaste. She may have acted that way because she thought such actions were permissible with an engaged person due to customs and traditions.