The word connection is good for talking about the way things relate to each other. Your special connection to your cousin might have something to do with your mutual love for science fiction and barbecue.
A connection can be physical, like the leash that provides the connection between you and your German shepherd, or emotional, like the connection you feel with your best friend. Any kind of direct relation is a connection as well, such as the connection a detective makes between a footprint at a crime scene and a suspect's favorite pair of shoes. The root is the Latin connexionem, "a binding or joining together."