Relationships certainly help in identity formation.
Our most important relationships are those with our immediate family. These relationships give us our most basic and lasting role identities: son, daughter, brother, sister, mother, father, etc.
Growing into adulthood, relationships continue to help define a person's "identity". Being friends with doctors and lawyers can shape the way you think of yourself just as being friends with actors and musicians can.
The fact that our relationships help to define us is clear, I think, in the construction of our values, our language, our social and political perspectives etc.
However, if we had to ask the question "What percentage of a person's identity is defined by relations, interactions, and connections to others in a social network?" the number is not going to be 100%. The percentage may, however, be very high.
Certainly our relationships help to define our self-image- we tend to compare and measure ourselves by the people that are around us, and many people feel that the extent to which they are able to fulfill their obligations to the people they are in relationships with is a big part of their worth as a human being. It is also true that we tend to define ourselves by the larger relationships in our lives- the communities we join, whether they are churches, political parties, teams we root for, etc. These are relationships too.
Yes, our relationships with others do help us define who we are. Our friends can bring out both the best and the worst in us. We realize what our faults are and what are our strengths. We learn to see the world from a different perspective. There is so much that we learn about ourselves through our relationships with others.
Relationships can reveal much about a person's character, in terms of kindness, loyalty, patience, honesty, etc.
There is a great poem called "Identity" by Julio Noboa Polanco, in which the speaker compares himself to an ugly weed among beautiful flowers. The poem beautifully addresses your statement about relationships.