Do adopted children have the same inheritance rights as biological children in Islam?

No — under classical Islamic law, an adopted child does not have automatic fixed inheritance rights the way a biological child does, because Islam does not recognize legal adoption in the sense of fully replacing a child's lineage (nasab); the child retains their biological identity and lineage even while being raised, loved, and supported by adoptive parents (kafala). This surprises many people, especially in cultures where legal adoption is treated as equivalent to biological parenthood, but it reflects a deliberate distinction in Islamic family law between raising and caring for a child (which is highly encouraged and rewarded) and altering inheritance lineage (which is not permitted).

This does not mean an adopted child is left with nothing — the adoptive parent can and is generally encouraged to provide generously for the child during their lifetime through gifts (hiba), and specifically through the discretionary one-third wasiyyah (see Q3 and Q11), since the child is not a fixed heir who would otherwise be excluded from a bequest. Muslim families raising adopted children, especially in India where this also has civil-law implications, are strongly advised to formalize this kind of provision (a will and, where relevant, gifts made during life) rather than assume the child will automatically inherit.

Source: Quran 33:4-5 (on adoption not changing lineage); general Sunni fiqh on adopted children and the recommended use of wasiyyah/hiba to provide for them.