Is it obligatory for someone who has paid Zakat al-Fitr, or money to charitable organizations, with the intention that this is on behalf of so-and-so and that is on behalf of so-and-so, and then the organization disburses it at its appointed time, to pay it again?

It is permissible to appoint an agent for the distribution of Zakat, whether it be Zakat on wealth or Zakat al-Fitr. If the giver of Zakat (muzakki) intends it as Zakat when handing it over to the agent, this is sufficient according to some scholars, and it is the most accurate opinion, even if the intention of Zakat is not present during the distribution by the agent.