Is mentioning the mother's name in official documents, and the procedures resulting therefrom, unlawful or a form of affiliation to other than the father?

If the mother's second family name is known, this is not considered a forbidden lineage. It is like saying, "I am so-and-so, whose father's family is such-and-such, and whose mother's family is such-and-such." There is no prohibition in it, and one does not sin by writing it.