Is a person considered to be from the Household of the Prophet (Ahl al-Bayt) if only their mother is a descendant of Zain al-Abidin ibn al-Hussein, knowing that their father is not from them, and is it permissible for them to inform people of this to mention Allah's grace?

It is not permissible for a man to abandon his lineage to his father and attribute himself to his mother just because she is from the Household of the Prophet (Ahl al-Bayt). The children of daughters do not enter into the lineage of Hashim. Al-Hasan (definition) and Al-Husayn are from the Household of the Prophet because their father, Ali, was the Prophet's cousin, not because they were the sons of Fatimah. There is no harm in informing people of this honor, as it falls under the category of proclaiming God's blessings, not out of arrogance or conceit.