How can one know if they are from the Household of the Prophet (Ahl al-Bayt) or not, and is there a source to verify someone's lineage, especially given that zakat is not permissible for them?

The Household of the Prophet (Ahl al-Bayt) to whom Zakat (definition) is not permissible are Banu Hashim and Banu al-Muttalib. This is the preponderant view according to Al-Shafi'i and some scholars, based on the Prophet's (peace be upon him) saying: "Indeed, Banu al-Muttalib and Banu Hashim are one and the same." The wives of the Prophet (peace be upon him) are also included among them. Their kinship is identified by the family or tribe's affiliation to this lineage through written documents or ancient inherited possession. It has been established among scholars that people are to be believed regarding their lineages.