Is it permissible to delay changing the name to that of the biological father in some documents or all of them, and what is the importance of lineage to one's fathers as commanded by Allah and His Messenger?

It is not permissible to delay changing the attribution of a name to the real father; rather, one must hasten to correct this, because it is forbidden to be attributed to anyone other than one's father. Allah the Almighty says: "Call them by [the names of] their fathers; it is more just in the sight of Allah." And the Prophet, peace and blessings be upon him, said: "There is no man who claims to be the son of other than his father, while knowing who his father is, except that he has disbelieved." Correcting lineage is important because it removes falsehood, preserves rights such as inheritance, and prevents the mixing of lineages, which could lead to incestuous marriages.