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In Texas, informal adoptions like informal marriages are recognized. 

A formal adoption is where the court system is involved.  A suit for adoption is filed.  The state investigates the adopting parents to make sure that they are good candidates for adoption.  The home will also be inspected to make sure that the adopted child will be provided for.  Once a report is filed, the judge approves the adoption and the adopted person becomes the child of the adopting parents with the same rights as a natural born child.  That means that the adopted child inherits not only from the adopting parents but through them.  They will inherit from grandparents and aunts and uncles to the same degree that a natural born child would inherit from them.

An informal adoption is similar to a common law marriage in that no formal legal proceedings are involved.  These adoptions may be called adoption by estoppel, equitable adoption or adoption status by the courts discussing them.  The courts treat these as contract cases in that once the facts are established, the heirs of the adopting parent are estopped from denying the adopted child his inheritance rights.  The courts look for an enforceable agreement between the adopting parent and the adopted child, his parents or some other person in loco parentis with the child.  The adopting parent agrees to adopt the adopted child then confers affection and benefits upon them and the person who agreed with the adopting parent relies upon the adopted status.

These informal adoptions are important where the adopting parent dies without a will.  If the adopting parent leaves a will, he can leave his property to anyone he chooses.  He can leave property to the adopted child or not.  However, if he dies intestate (without a will) then the child adopted by estoppel will inherit a share of this adoptive parent's estate.  

The informally adopted child will not inherit from his adoptive parent's parents unless they were in privity with the adoption.  He will only inherit from his adoptive parents not through them.  In other words, persons not in privity with the adoption are not bound by the adoption. 

Example:   There is an adoption by estoppel where Joe adopts Jane. Joe dies without a will.  Jane inherits all or part of Joe's estate the same as any child would.

Example:  There is an adoption by estoppel where Joe adopts Jane.  Joe dies with a will that leaves his estate to his parents who survive him.  Jane doesn't inherit from Joe because of the will.  Later, Joe's parents die without a will.  Jane isn't entitled to inherit from Joe's parents because they were not in privity with the adoption.