What is the nature of the relationship between a wife's sister (brother's wife) and a brother in Islam? Is it permissible to call the wife's sister "my sister" and treat her as a sister?

It is not permissible to treat one's brother's wife as a sister because she is not a mahram. Therefore, it is not permissible for her to appear before you without the shar'i hijab, nor is it permissible to be alone with her (khalwa), or to shake her hand, or to look at her, or to speak with her except from behind a veil if there is no fear of fitna. This is because Allah has forbidden believing women from displaying their adornment except to specific individuals, as He says: "And let them not display their adornment except to their husbands or their fathers..." (Qur'an 24:31).