The "schizo-" in the term "schizophrenic" refers to a split from reality - not a split personality. Although there are schizophrenic people with a split personality (multiple-personality disorder) it is exceedingly rare (about one-in-ten million cases of schizophrenia). 

When we speak of a schizophrenic being split from reality, we mean that their perceptions are distorted by their cognitive processes to the extent that they are not in touch with reality any longer. For example, someone who is angry because they believe someone has done something which they have not actually done (a paranoid schizophrenic who believes that someone is working in league with space aliens and tries to kill that person in imagined "self-defense"). That's an extreme example, but you get the general idea?