Is it considered usury for a father to take Eid money from his son, promising to return it with an increase as a reward, or for a ruler to request his subjects to deposit their money with him, promising to return it with an increase as a reward?

What the questioner mentioned is neither a sale nor an exchange, but rather a preservation and a reward from the father to his son, as he preserved his money for him and increased it as a reward. However, if it were a loan or an exchange, then it would be Riba (usury/interest). Most scholars hold the view that Riba applies between a father and his son, because the son's money is his own property. Some contemporary scholars have disagreed with this, but without acceptable jurisprudential support.