If they took the car in and created an appointment for it (there is an appointment in your app), its likely that someone needs to close the "ticket". Note, I dont work in the car industry, I work in the IT industry, but I am 100% sure that tesla uses an intake system that assigns technicans to work and documents it, just like in any sizable company there is an IT system to track inventory, and for end users to report problems and submit "work requests".
If your app still says "pick up", its likely that whoever checked you out did not close out the work order. I would start there (by asking the service center if they see a work order for your car, and if so, could they please close it out).