Rei Ayanami and Shinji Ikari are supposed to be brother and sister. Why do they not both have the same surname as their father, Gendo Ikari?
They are not supposed to be brother and sister. Rei was created using material of Yui Ikari which would make their relationship closer to mother/son or aunt/nephew genetically speaking and this origin was kept secret.
Gendō Ikari brought his first clone (Rei I) to the Gehirn base, which is the predecessor of NERV and responsible for the development and construction of the first Evangelion units, introducing her as the daughter that an acquaintance entrusted to him.
Officially, Rei II is chosen by an organization called the Marduk Institute as the First Child and pilot of EVA Unit-00, though this organization is later revealed to be a front, with Gendō and the SEELE council pulling all the strings.
So officially there is no relation between Gendo and Rei except that he is responsible for her. Giving her his surname would serve no purpose and instead make others more suspicious of her and what connection she shares with him.
Quote from Wikipedia