As Josh pointed out everyone is born with a fraternal twin (barring accident or illness).
It would seem that you would have as strong culture around twins and what you twin does, whether this is conforming or rebelling against what the other is doing. The society would probably have a special name for those who have lost their twin, maybe being a "single" person is a stigma there.
It would also seem not uncommon for double twin marriages ie two twin siblings to marry two twins that are also siblings.
Double marriages would tend to create tightly knit extended families because when families are united in marriage they are actually united in double marriage.
It might also create a tradition of conformity, you grow up you twin who is like you, your parents, you siblings and their twins, you parent's twins who are like them and your parents' twins children who are also like you.
They would also limit the mixing of genes. In our world a if two children get married they usually marry into different families and so mix genes with two different families, in this model they would only mix with one more.
Finally it limits control over birth rates, unless you abort one child in a twin pair the only options are 0 kids 2 4 6 .... Aborting one child in a twin pair might stigmatize the survivor and so be unlikely, unless there was some movement to make it popular.