Old Persian does not have "family names" the way you think of it. In the XPh description Xerxes begins:

I am Xerxes, great king...

Adam Khshayarsha, Khshayathiya Vazraka...

He goes on to list three more royal tites, then

... King Darius' son, an Achaemenian, a Persian, the son of a Persian, an Aryan, of Aryan heritage

Dârayavahauš \ xšâyathiyahyâ \ puça\ Haxâmanišiya \ Pârsa \ Pârsahyâ \ puça\ Ariya \ Ariyaciça \ thâtiy\

Darius similarly lists his titles, then:

Vishtaspa's son, Arshama's grandson, of Achaemenian heritage

Vištâspahyâ \ puça \ Aršâmahyâ \napâ \ Haxâmanišiya \ thâtiy

So we can list several identifiers:

-Patronym in the form of (father)'s son

-Clan (Achaemenian)

-Tribe (Persian)

-Ethnicity (Aryan)

I've named the last two identifiers a bit arbitrarily, but you get the idea. Depending on who you were talking to, you'd identify yourself differently. In principle it was possible to grammatically construct this in several ways, just like how you could say "the Persian Darius" or "Darius the Persian". For a notable example, in the Avestan Gathas, the author identifies himself as "Spitama Zaraθustra", i.e. Zaraθustra of the Spitanan clan.

I hope this gave you what you were looking for? I can take followup questions if needed.