How is the son of zina treated in society when he does not possess official documents, given that he is not attributed to his father?

A child born out of wedlock is not attributed to his biological father from adultery due to the illegality of such a lineage. However, he can be attributed to a name that signifies servitude to Allah, such as 'Abdullah ibn 'Abd al-Rahman (Servant of Allah, son of the Most Merciful), and this can be officially registered in formal documents.