Why am I so bothered by the behaviour of a younger cousin and how do I get over it?

There is an old saying, "Imitation is the greatest form of flattery." She looks up to you and wants to copy you. You can still be your own person and, believe me, this is just a phase she is going through. She will eventually find her own style and develop a stronger personality than she has now. You are lucky to have family live so close to you.