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

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

Use kinship, because you are referring to a

relationship: the way in which two or more people feel and behave towards each other

Cambridge

in particular, you refer to a family relationship. This is known as

kinship = the relationship between members of the same family

Cambridge

The word comes from kin, which is:

a group of persons of common ancestry

Merriam Wenster

And the suffix -ship

= -ship state : condition : quality. Example:friendship

Merriam Webster

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.