It depends on the reasons your psychiatrist is saying this to you. She may have specific reasons to believe that you are ok. But without seeing you face to face for an evaluation, there is no way for me to know all of your symptoms, your history and treatment which is important information to predict how you might do in the future. However, it may be more helpful to you to ask your doctor why she believes that you can fully recover. She can better explain it and give you the reasons why she feels this is true.
You are right, anyone can have mental health issues, even with no history. It all depends on the stresses, family history and other factors. People are not predictable and things can happen. But with the right support and treatment, most people can be helped.
Schizophrenia is usually not diagnosed until late teens or early 20. And you may have not had adequate access to services as a child or poor care that resulted in a missed diagnosis for the OCD, which can be diagnosed in children.
I hope this has answered both of your questions.
Schizophrenia is only now being diagnosed in children. It probably was not when you were young. And even now, it is not that common.
You could not have been diagnosed as a child with Schizophrenia since it was not a diagnosis back then.
I hope this has answered your original and new question. You may need to close this thread and start a new one if you have further questions.