Is it possible for one "person" to posses both female and male genitals?

Its possible. Genitalia development depends on the chromosomes and exposure to testosterone. There is a sequence of events starting with the internal genitalia (ovaries, tubes, testes) that ultimately play a role in the development of the external genitalia (outer vagina or penis). A female could have an enlarged clitoris that looks like a penis. Others could have part ovary and testis.