When grandparents adopt their own grandchild it is usually the result of extremely difficult circumstances involving the child's parents. Whether the child's parents are unfit, have abandoned the child, or are deceased, helping the child have the right perspective on the circumstances will be challenging. I believe the healthiest way to see the role of such grandparents is to say that it has been expanded, rather than changed.
Children can adjust to living in situations other than those in which there are two happily married, biological, parents; but it is important to help a child of a "non-traditional" family see his/her situation as "normal". It is also vitally important to find a way to instill a sense of real security and stability in the child.
Whenever someone adopts a child that person is, of course, the child's parent. Being an adoptive parent, however, also means finding a way to help the child know the truth about his/her beginnings while also being able to have a healthy and appropriate perspective. The same truth can be presented in different ways, and those different ways can send different messages. For example, an adoptive parent can send the message, "Poor you. Your mommy has serious problems, so I had to adopt you. I wish things were different for you." A different message sent to the same child could be, "Your mommy, whom I love very much, was not able to take care of you; but she knew how much I love her and you, and knew you and I belong together."
It is generally agreed that adopted individuals benefit when they know their roots and have some idea about the circumstances under which they were placed for adoption, so being honest with the child about being his/her grandparent would seem to be a good way to start presenting the truth. Not telling the child about the adoption is generally not considered the healthiest approach. If the biological mother remains in life of her parents and child it could potentially be confusing for a child if all the adults involved try expect to be seen as his parents.
Whether in the role of biological parent, adoptive parent or grandparent, the loving adult who raises the child, bonds with the child, and takes on parental responsibilities is obviously acting as a parent. The legal adoption makes it official that the person is a parent. What that person calls herself, though, may be the only question. This is where, I believe, it is important to reinforce for the child that there is continuity in his/her life, and that the role of his/her grandparent has been expanded, rather than changed.
I believe that explaining to the child who his/her biological mother is, what the role of a parent is, and how adoption makes someone other than the biological mother a child's parent would be the way to handle the situation. It would seem to me that a child would feel most secure and understand the situation best if the grandparent said something like, "Susie is your biological mother. I am your grandmother. That's why you call me, 'Grammy'. When I adopted you it meant that I also became your parent forever because Susie is not able to be a parent."
When it comes to grandparents, in healthy situations grandparents and grandchildren are extraordinarily close, whether or not children are with their biological parents. Generally, the bond between grandparents and grandchildren is a powerful one that should not be underestimated. When children are cared for by grandparents (even in situations not involving being adopted by them) they are closer still. I believe this relationship is an important enough one that nobody should decide to sacrifice it in favor of being a child's parent. It seems to me that a child could have a better perspective by seeing his/her grandparent in an expanded role, rather than by being faced with having to sort out the differences between mothers, adoptive mothers, grandmothers, and other grandmothers.
The definition of the word, "grandparent," is related to the individual's relationship with one of the child's parents. The definition of the word is not derived from whether or not an individual indulges the child, although many people have come to associate that connotation with the word. Not all grandparents indulge children more than the children's parents do. Not all grandparents spend more "special time" with children than the children's parents do. Parents and grandparents all have different types of relationships with children. In the "standard" situation in which a child has parents, and grandparents have the traditional grandparent role, it is not appropriate for grandparents to take on the general parenting role. Good parenting, however, is something that comes for love, care, emotional maturity, and common sense. Good parenting includes offering a child what s/he needs, and I suspect that most grandparents who love their grandchild enough to adopt him/her will naturally lean toward being the good parent that the child needs.
Keeping the title, "grandparent", while assuming the additional role of parent is not keeping a title that implies inferiority. When grandparents have been solid, loving, good, parents to their own children they don't lose their parenting skills just because their children have children. When they have made mistakes with their children they have the benefit of hindsight to have learned from those mistakes. Good, solid, loving, grandparents don't usually have fewer parenting skills than their children do because they have the benefit of experience, wisdom and the perspective of maturity, in addition to the parenting skills they've always possessed. When a grandparent takes on the additional role of parent s/he should not see the title, "grandparent", as "less"; and s/he should not send the message to the child that the title means having a parent who is "less". I believe that the grandparent who adopts a grandchild should proudly make it clear that s/he is someone who is clearly "more" or "both".
One of the best examples any parent can show a child is that people can, and should, define their own role, regardless of any labels placed on that role. Another important example a parent can show a child is that responsible, loving, adults rise to the occasion, adapt, and find ways to offer the child they love what every child deserves - good parenting regardless of the title of the person providing it.
When a person adopts his/her grandchild that person is both the child's grandparent and parent. It is a fairly common situation and not very difficult for a child to understand. Helping the child see that grandparents can be parents too may be the best way for the child to understand the truth. More importantly, it could be one way to help a child see the stability and continuity offered by his birth family.
The world is full of adults who have become very successful and famous, and who have stated that they owe it all to their grandmother, who raised them, taught them right from wrong, and was "both mother and father" to them as well. These are people who have seen for themselves all the things a grandparent can be without being required to sacrifice the title, "grandparent."