Do you need a license to own a owl?

Not in the United States. They are strongly protected as a "bird of prey" by National laws. You cannot own, buy, sell, injure or harass an owl. By special permit, some professionals are able to keep them, but it is difficult to obtain these permits.

The primary reason is that Birds of Prey help keep rodent (mice,rats) populations down.

Even Native American tribes must apply for permits to use eagle feathers in their ceremonies.