Does the official registration of a name according to the state system (real name - mother's family name - father's family name) or (real name - mother's father's name - father's name) fall under the severe warning against attributing oneself to other than one's father?

A newborn must be attributed to his father and no one else, as Allah the Almighty says: (Call them by their fathers' names; it is more just in the sight of Allah). The Prophet, peace and blessings be upon him, forbade a man from claiming someone other than his father while knowing [that fact]. This means attributing oneself to him and taking him as a father. The wisdom behind this prohibition is to avoid the mixing of lineages, the violation of sacred norms, and the disruption of inheritance rights.

As for the first form of writing the name (the newborn's name, then his mother's family name, then his father's family name), if it is mandated by the state, it does not fall under the warning against attributing oneself to other than the father, because it does not nullify attribution to the father. People commonly understand that what is written is the family name, and this is a designation in documents that is not usually relied upon to ascertain lineage.

As for the second method (person's name, then mother's father's name, then father's name), it does not fulfill the Sharia objective. If it is possible for a Muslim to write his name and his father's name according to the customary practice known to Muslims without harm, then that is better.