Think of it this way:
* Undergraduate refers to the *status* of a student who is pursuing a degree but hasn't yet completed it.
* Undergraduate degree refers to the *degree itself*, which is awarded upon successful completion of undergraduate studies.
The name reflects the student's position in the overall academic hierarchy. Once you complete your undergraduate studies, you are then eligible to pursue postgraduate (or graduate) studies. The undergraduate degree signifies the completion of the first major phase of higher education.