Adopting a child with their consent and raising them is permissible according to Shariah. However, changing the adopted child's father's name is considered a grave sin. That being said, it is permissible to list the adoptive parent as the child's guardian.
Furthermore, an adopted child does not inherit from their adoptive parents; they are only entitled to a share of their biological parents' inheritance.
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دلائل:
القرآن الکریم: (الاحزاب، الایۃ: 5)
اُدْعُوْھُمْ لِاٰبَائِہِمْ هُوَ اَقْسَطُ عِنْدَ اللّٰہِ․۔۔۔۔الخ
صحیح البخاری: (باب من ادعی الی غیر ابیہ، رقم الحدیث: 6766، 273/6، ط: دار الکتب العلمیة)
عَنْ سَعْدٍ رَضِيَ اللَّهُ عَنْهُ ، قَالَ : سَمِعْتُ النَّبِيَّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ، يَقُولُ: مَنِ ادَّعَى إِلَى غَيْرِ أَبِيهِ ، وَهُوَ يَعْلَمُ أَنَّهُ غَيْرُ أَبِيهِ ، فَالْجَنَّةُ عَلَيْهِ حَرَامٌ.
Allah the Almighty knows best.
Darul ifta Al-Ikhlas, Karachi.