This question can be answered from a couple of different perspectives. You could do a bit of research to find actual studies and statistics to support the claim that family does indeed develop identity and provide a sense of belonging. Or, you could simply speak from personal experience and observation.
Family psychology is a relatively contemporary study of behavior. The idea that individuals are highly influenced by their immediate and extended family (and the "family environment") is based on the principles of the systems theory. In this, the family is considered a system.
If you look at this question from a more clinical perspective, it is easiest to define a family's influence on individual behavior by looking at negative behaviors. Use the example of a child growing up in a broken or single-parent home where the environment lacks a sense of support, supervision, and security. This is actually a pretty common occurrence in today's society, and family psychologists would now assert that there is a direct correlation in the behavior of children who display learning and social difficulty in school (at all age ranges) and the home and family environment.
Looking at this from another perspective, there are numerous studies and statistics showing the difference in the success rate of children who were made to sit down to a "family dinner" on a regular basis while growing up. A regular family dinner a person's background is linked to higher educational success, a lowered risk of teen pregnancy, a lowered risk of drug and alcohol abuse, a lower risk in childhood obesity, and a general increase overall physical and mental health.
From a more personal perspective, I could tell you about growing up in a large family where my siblings and parents were all very outgoing, successful, healthy, and supportive of one another. In our adult lives, all of my siblings and I have maintained jobs in our first career choices, married spouses whom we still love, and are raising mostly happy and healthy children. We would all agree that our current sense of happiness is due to the stability of our house and family while we were growing up. Our parents are still married to this day, and as a result we have learned that love and commitment are worth fighting for. Because there were four of us, we learned very quickly that we were not (and could not be) the center of attention all the time. As a result, each of us worked hard to become independent, but always felt a strong sense of support from each other and our parents.
Finally, we were raised in a Christian household that regularly attended church. Though each of us has had a period in our adult lives of straying from this habit, it seems that as each of us raises our own families, we are all going back to the faith we knew growing up. Not only do we each have a very strong sense of identity and belonging in our immediate family, but we each have a strong sense of identity and belonging in a spiritual family as well. The sense of security both provided to us as children is something we all wish to provide to our own children now.