Are my sister and her husband considered to be living in a state of sin, especially after he uttered the phrase "you are divorced" multiple times, including three clear instances, and each time he tells her "you are divorced," even after the three confirmed divorces?

If a man divorces his wife three times, she becomes irrevocably separated from him (baynunah kubra). She is not lawful for him until she marries another husband, based on the Almighty's saying: "And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband." It is forbidden for him to have intercourse with her or to enable her to do so. She must escape from him by fleeing, or by ransoming herself with money, or by taking the matter to court. Marriage is a strong bond, and the matter of divorce should not be taken lightly. The شرعية (legal/religious) principle is to hold together honorably or separate kindly. It is preferable to avoid divorce as much as possible, and for understanding and respect to prevail between the spouses.