I feel like many of the people claiming that there's no reason to be offended because they have mental illness in the family do not understand the severity of schizophrenia (I am sure this does not apply to all, but some). Schizophrenia is not anxiety, it is not depression, it is not ADHD. Schizophrenia is more often than not a COMPLETELY debilitating disease resulting in the total loss of a person - first their actual identity as a functional member of society and then often dying alone, sometimes alone on the streets, to suicide, to drug addiction. The paranoia is all encompassing and leads to a life of cycling between psych wards and prisons. They are often severely dangerous to people around them, causing even family members who love and want to help them to have to ditch them to protect themselves.
There may be avenues of dark humor to joke about schizophrenia, but Disney, between cast members and patrons, is not one of them. Do you also think it would be "too sensitive" to be offended by a joke about cancer? About suicide? How about obesity? Can CMs make jokes about obesity? Racist jokes? Making fun of gay people? Mentally handicapped?
Those things seem more obvious that a cast member shouldn't joke about them. What makes schizophrenia any different? Probably because you don't understand the condition, don't know anyone with the condition, and don't have to deal with the agonizing and ongoing loss of a loved one to that condition. Most people's day to day interaction with schizophrenia extends no further than crossing the street to walk avoid walking past the loud, exceptionally dirty, "scary" homeless person. That's the face of schizophrenia, and there is someone who loves or loved that person once. It isn't funny.
Besides that, just to reiterate what others have said, the statement is inaccurate about what schizophrenia even is. Multiple personality disorder is not schizophrenia.
