Is the questioner considered a lawful wife to that young man who informally contracted marriage with her then abandoned her?

Marriage without a wali (guardian) and with an agreement to conceal it is zina (fornication) and not a lawful marriage. The wise Sharia prohibited women from immodest display (tabarruj), reckless mixing, and speaking with a non-mahram man without necessity. It commanded them to observe hijab (veiling) and modesty, and forbade them from touching or being alone with a non-mahram man. All of this is to protect their chastity and society. The Prophet, peace be upon him, ruled that a woman's marriage without the permission of her wali is void, stating: "Any woman who marries without the permission of her wali, her marriage is void, her marriage is void, her marriage is void." This is what the scholars have unanimously agreed upon. Therefore, a secret marriage that is agreed to be concealed is void and akin to illicit sexual relations (sifah). Thus, what transpired between the questioner and the man is not a lawful marriage. She must repent to Allah, regret what has passed, and resolve not to return to such an act.