Is a reciprocal relationship based on affection and trust, which includes the exchange of secrets and reconciliation upon disagreement between a man and his brother's wife, considered permissible (halal) or forbidden (haram) in Islam?

The brother's wife is a stranger [non-mahram], and it is not permissible to be in seclusion with her or to look at what is unlawful of her. The fact that she is your brother's wife makes the matter more serious, for "the in-law is death" [a reference to a Hadith warning against being alone with the husband's brother]. Therefore, it is obligatory to avoid seclusion with her and to refrain from prolonged conversation with her, except when necessary.