Does the dispatcher belong to an airline or an airport?

The dispatcher belongs to the airline.

The dispatcher works in the operation control department of the airline. They are responsible for collecting flight information, making and applying for flight plans, and publishing each flight with the captain. Dispatch is one of the core departments of airlines, which has the decision-making power to postpone, allocate or even cancel flights. Each flight needs to be signed and released by the dispatcher, and relevant information such as release form, flight plan, weather situation and forecast, navigation notice and minimum equipment list shall be provided. Dispatchers are also responsible for handling air emergencies and completing other tasks assigned by leaders.