Is it Islamically permissible to register the apartment in the daughter's name after its price has been paid in full, given that it was initially registered in the father's name due to the initial payment circumstances?

It is permissible for a father to gift a house to his daughter, and it becomes her property upon her taking possession and control of it. For the gift to be effective, the majority of jurists stipulate that the house must be vacated of the giver's belongings. The Malikis stipulate that the giver must not reside in it after the gift, while the Hanafis permit the giver to reside in it if it is occupied by his belongings. The Shafi'is hold that the gift is invalid if the house is occupied by the belongings of someone other than the donee. However, if the father intends to make the gift effective upon his death, it is a will, which is not permissible for an heir or for more than one-third of the estate, unless approved by the heirs. The father must treat his children fairly in giving gifts, unless there is a valid reason to favor one of them, and he must not intend to harm the heirs.