Is a woman considered legally a wife merely by a man saying "you are my wife" and her agreeing to it online? Is it permissible for her to marry another man? How can she separate from him if she has become his wife? And what should she do to gain Allah's forgiveness?

The questioner erred by chatting with the young man online, as this is impermissible and leads to much corruption. What transpired between you is not a legitimate marriage, for marriage has conditions without which it is not valid, and among the most important of these are the presence of a guardian and witnesses. A man speaking to a woman and telling her "you are my wife" is not a legitimate marriage, and it is not permissible for him to have intercourse with her based on that. You have the right to marry someone else. One who does this cannot be a student of Islamic knowledge or a memorizer of the Book of Allah. What is obligatory upon you is to repent and cut off the relationship with him. The hadith: "There are three matters whose seriousness is serious and whose jesting is serious: marriage, divorce, and return [to a marriage]" pertains to a marriage that has fulfilled the conditions of validity, and what occurred between you is not a marriage at all.