TV productions are a product, not a service. Apple TV is the service.

Apple has a tv service and Apple also has exclusive content, which they brand with “Apple TV”…so it’s kind of both.

Same for the other big streaming services. Some of them (Netflix, Prime Video) are more involved in content production, up to and including having production facilities and an in house staff. But a lot of the “exclusive” branded content is made by semi-independent production companies.