Actually you can have a ticket applied to more than one person in an agile process. You must've been thinking of Scrum in particular which does not allow the practice.

(The original agile process called Extreme Programming actually mandated pairs.)

Indeed, a task not budgeted in time is not done. That pertains to tests, static quality analysis, performance optimizations and use case analysis. None of these are explicitly budgeted in agile processes, though they say "write tests" or "definition of done", these are taken as suggestions.