Praise be to Allah.
The stepdaughter is the daughter of a wife from someone other than her current husband; she is permanently forbidden in marriage to the man if he has consummated his marriage with her mother. This means that she has become one of his mahrams.
It says in Fataawa al-Lajnah ad-Daa’imah (17/367):
If a man marries a woman and consummates the marriage with her, it becomes permanently forbidden for him to marry one of her daughters or one of the daughters of her sons, no matter how far the line of descent extents, whether she is a daughter from a previous or subsequent husband, because Allah, may He be glorified and exalted) says (interpretation of the meaning):
“Prohibited to you [for marriage] are your mothers… and your step-daughters under your guardianship [born] of your wives unto whom you have gone in”
The stepdaughter here is the wife’s daughter. A man is regarded as a mahram of the daughters of a woman whom he marries and with whom he consummates the marriage; it becomes permissible for them not to observe hijab in front of him. End quote.
With regard to the rights and duties of a stepdaughter and her mother’s husband towards one another, they may be summed up as upholding ties, respecting one another, and treating one another kindly. All Muslims are enjoined to treat their fellow Muslims with kindness, so how about those who become mahrams due to ties through marriage. Undoubtedly they have a greater right to kindness and care than Muslims in general.
But spending on maintenance, serving and obeying are not obligatory between them. The stepdaughter does not come under the same ruling as her mother in these matters in terms of it being obligatory according to Islamic teachings. If the husband spends on his stepdaughter out of kindness, and she responds by treating him well, taking care of his house and serving him, then that is good, because bringing people together and creating harmony between them is an aim that Islam seeks to achieve.
The husband should understand that kind treatment of his wife includes kind treatment of her daughter. And the daughter should understand that part of honouring her mother is honouring the mother’s husband and treating him kindly.
Shaykh Ibn Baaz (may Allah have mercy on him) said: (25/365):
There is the hope for the one who takes care of women other than his daughters, such as his sisters, paternal aunts, maternal aunts, and others who are in need, treating them with kindness, and providing them with food, drink and clothing, will attain a reward like that mentioned by the Prophet (blessings and peace of Allah be upon him) with regard to the one who takes care of three daughters, for the grace and mercy of Allah are immense. The same applies to one who takes care of one or two daughters or others, and treats them kindly; there is the hope that he will attain a great reward, as is indicated by the general meaning of the verses and hadiths that speak of kindness towards the poor and needy, relatives and others. If this is the virtue of showing kindness towards daughters, then showing kindness towards parents, or one of them, or grandfathers or grandmothers ,will bring an even greater reward, because of the great significance of parents’ rights, and the obligation to honour them and treat them kindly. It makes no difference whether the one who is showing kindness is a father or a mother or anyone else, because the reward is connected to the deed. And Allah is the source of strength. End quote.
The Standing Committee for Issuing Fatwas was asked (25/296):
How should the social family ties be in the Muslim household?
Allah has enjoined upholding that which will strengthen ties between family members. He enjoins upholding ties of kinship, and treating relatives kindly, as He, may He be glorified, says (interpretation of the meaning):
“And fear Allah, through whom you ask one another, and the wombs. Indeed Allah is ever, over you, an Observer”
“Worship Allah and associate nothing with Him, and to parents do good, and to relatives…”
“Say, Come, I will recite what your Lord has prohibited to you. [He commands] that you not associate anything with Him, and to parents, good treatment, and do not kill your children out of poverty; We will provide for you and them”
“And your Lord has decreed that you not worship except Him, and to parents, good treatment”
And there are other similar verses in the Qur’an.
It is proven that the Prophet (blessings and peace of Allah be upon him) said: “No one who severs ties of kinship will enter Paradise.” Narrated by al-Bukhaari and Muslim.
And he said: “Whoever would like to his provision to be increased and his life to be extended, let him uphold ties of kinship.” Narrated by at-Tirmidhi. And he said: “ Allah has forbidden to you disobedience towards mothers and burying daughters alive…” Narrated by al-Bukhaari and Muslim. And there are many other hadiths which encourage upholding of ties of kinship, adhering to Islamic etiquette and noble characteristics, and maintaining kind treatment, all of which strengthen the bonds between families and family members, and bring all Muslims together. This is the right way, not drifting away from the path of Islam and ignoring Islamic etiquette and the call for good manners. End quote.
And Allah knows best.