Is this brotherhood between a man and a woman, who are not related by blood or suckling, and which reaches the extent of kissing between them and treating each other completely as siblings in what is permissible and what is forbidden, Islamically permissible?

What you mentioned in your question has no relation to legitimate brotherhood; rather, it is reprehensible and contrary to Islamic law. There are legal boundaries between a man and an unrelated woman that must be observed. What is obligatory is to repent to Allah and to interact with women in accordance with Islamic regulations, as a precaution against the footsteps of Satan, who enjoins immorality and wrongdoing.