Soul. The identity of the soul contained in Eva-00 is never directly stated, but analysis of various clues and hints dropped through the series points to it being the soul of Rei 1, the first Rei Ayanami clone (or a part thereof). For more in-depth analysis, see "Eva-00's soul".

Does unit 0 have a soul?

The true nature of the soul inside Unit-00 is never specified, but it is implied to be directly connected to Rei in some fashion. Unit-00 was self-destructed by Rei in order to destroy the sixteenth Angel, Armisael.

Is Unit 1 Shinji's mom?

Shinji's mother Yui died during the Contact Experiment, and her soul was fused into the body of the fleshy prototype Unit-01 when her body was destroyed. Asuka's EVA contains the maternal part of the soul of her mother, who went crazy as a result of her Contact Experiment and committed suicide.