Is raising a sponsored orphan girl to call the couple "father and mother" and their relatives "grandfather and grandmother, paternal uncle and aunt, and maternal uncle and aunt" considered an attempt to affiliate her to them, and is this permissible for them and their relatives?

There is no objection to accustoming the fostered child to call his foster parents "father" and "mother," and for them to call him "my son," if this is done out of respect and affection, provided that this does not lead the fostered child to believe that the foster parents are his biological parents.