Personality is not fixed. Your personality adapts depending on the situation you find yourself in. This is completely normal. Even without thinking about it, you will act differently depending on the situation.
Emotions are the same way. It is not always appropriate to show certain emotions. For example, you wouldn’t want to show how angry or upset you are during a work meeting with an important client. You wouldn’t want to start laughing at a funny joke while at a funeral.
You may be a focused and organized person at work, but carefree and relaxed while out with friends. This doesn’t make you fake in one arena and your “true self” in another – you reflect different parts of yourself in different situations.
Impression management is similar, but there is more thought involved. You adapt yourself on purpose.