Is a mother sinful for writing an apartment in her daughter's name, knowing that she had designated it for her from the beginning, and the father and siblings are aware of this?

As long as the mother has been just in her giving among her children, then her registering the apartment in her daughter's name and creating a fictitious sale to establish that right is permissible, provided that the daughter takes possession of the apartment during her mother's lifetime to prove her ownership. Otherwise, this would be considered a bequest to an heir, which is not permissible unless approved by the other heirs. However, if justice is not observed among the children in their gifts, then it is forbidden to favor some of them without a valid reason, due to the Prophet's (peace be upon him) saying: "Fear Allah and be just among your children."