What is the ruling of Islamic law on sponsoring a sister's daughter for upbringing, and is it permissible for her to remain in the house after reaching puberty, and what is the ruling on that, especially after the couple becomes old?

Adoption—which means attributing a person to a lineage that is not his own son—is forbidden and invalid in Islam, and the rulings of legitimate sonship do not apply to it. However, taking in a relative and sponsoring them for companionship, without attributing them to one's lineage, is not forbidden. Nevertheless, the niece is a stranger to her aunt's husband, so she must observe hijab in front of him, and private seclusion with him is not permissible, unless the aunt had breastfed her sister's daughter with a mahram-creating suckling. In that case, her husband becomes a mahram to her, either because he is her father through suckling (if the milk was due to his intercourse), or because he is her mother's husband through suckling.