1. Holding two separate degrees: This refers to someone who has completed two distinct academic programs, earning two separate degrees. For example, they might have a Bachelor of Arts in English and a Bachelor of Science in Computer Science. These are usually pursued sequentially, but could sometimes be concurrent (though that's less common).
2. Graduating twice from the same institution: This is less common, but it could mean someone has graduated twice from the same university, perhaps with different majors or at different levels (e.g., undergraduate and then graduate degree). This is often associated with significant time lapse between graduations.
The context is crucial to understanding the meaning. If someone simply says "double graduate," clarifying which meaning they intend is important.