you are right in estimation: it could be dangerous that 'own' a task but not able to successfully produce the result! Your performance review will be lower-down!
your boss seems not a very reasonable person.
If you still want to stay in current position, my two cents are:
(1). negotiate with your boss for the task 'owernship' vs 'responsibility' issue.
(2). get in touch with a employment attorney, let them know your situation and help you, in case of worst situation, them know how to protect you at law point of views!