They can be a family, and any live in partner is automatically fiscally responsible under the law for any of their partners children anyways. However, I don't believe a child should have more than 2 primary parents legally. It is too complicated to subject a child to the court battles that will inevitably follow between 4 people instead of 2.  

Mature adults should be able to pick two primary parents, same sex or not, biological or not, to have primary legal rights for custody, which could revert to the two stand by secondary parents in event of death of a primary. Biological grand parents could be given the same legal standing as secondary parents. 

Four legal parents with equal rights is not what is best for children. And birth certificates should have birth parents on them only. Its a vital statistical document. Not even adoptive parents should be on birth certificates.