Is it permissible for a man to marry the daughter of a woman with whom he previously had a relationship that included all forms of fondling and direct physical contact without penetration, knowing that the old relationship ended five years ago, and that the mother now considers him like a son to her, and that the girl loves him and he loves her?

First, you must sincerely repent, frequently seek forgiveness, and perform good deeds, for Allah Almighty has enumerated and recorded all things. As for your question, you are permitted to marry this girl, and what you did with her mother does not affect the permissibility of your marriage to her. However, we do not advise you to marry her, especially if her mother has not repented and mended her ways, out of fear that Satan may beautify sin for you both, or that the daughter may discover what transpired between you and her mother, which could cause her distress and lack of trust, potentially leading her to disobedience, disrespect, or committing forbidden acts.