Is depression part of who a person is?
Depression can be defined as "a state of feeling sad" (Merriam-Webster), which is an emotion or a feeling in the heart. 

Depression can develop as a reaction to a thought, demand, event or word someone said to you.

You may have a certain "character" or "temper" that makes you vulnerable for things that hurt you and make you sad. So, it is correct to say that you are not depressed as a person but you may be prone to become depressed when something hits you.

Depression is a feeling and not a disease and it usually does not come from an underlying disease, but from the conflicts in your thoughts, attitudes, believes and relationships - which you can accept or change, as appropriate. Depression can make you stop functioning, which you can call a disease, but depression on its own is not already a disease.