What is the name of the relationship between someone and his/her relative?

A friendship is a relationship between people that may change over time as the friends come to know each other better.

A relationship with kin is a kinship, but it’s static, not dynamic. Kinship is a defined relationship like brother or cousin, and stays the same no matter how well you are acquainted. Different cultures define kinship relationships differently, especially, perhaps, what we call in-laws in English—relatives because of marriage rather than blood lines.

This is a family relationship, none are closer. So it would be appropriate to refer to it as your relationship with your cousin.