So if an alien abducted you, created a clone to replace you while you are experimented upon, you'd still agree that you are living normally, and nothing happened to "you"?
For example, at work I may keep to myself and be very quiet - in that setting my actions would define me as an introvert. With my friends I may be more outgoing and talkative, which is more characteristic of an extrovert. So I have two identities depending on my behaviour. I am not my introverted self when with friends, and not my extroverted self when at work.