Is it permissible to open Islamic bank accounts for children and deposit gifts, Eid allowances, and monthly sums into them, or must the money remain the property of the father to be distributed as an inheritance after his death?

Gifts to children are permissible on condition of fairness. However, if the ownership transfer is deferred until after death, it is a bequest, not a gift. A bequest to an heir is not permissible. As for a gift, the ownership transfer occurs immediately. The father takes possession of the gift on behalf of his two sons. If a father gifts something to his young son, no possession is needed; witnessing is sufficient. As for gifts or Eidiyat (Eid presents) from others, which you accepted on behalf of your child, he becomes the owner of them by that acceptance, and they are exclusively his during your lifetime and after your death. The wealth of each child, if it reaches the nisab (minimum taxable amount) and a hawl (full lunar year) passes over it, zakat becomes obligatory on it.