Are tasks in the calendar the same as in Peak Task, Day and Board?

Yes. A task exists once. The calendar shows it on its scheduled day (as a block if it has a time), otherwise on its due date. If you move or complete it here, it's moved or completed everywhere. Routines are recurring tasks from Peak Task; the next one is created when you check off the current one. Bookings from Peak Booking and scheduled sales sessions appear too; you change them where they come from.