If you work at a university and take courses, is the degree the same?
Many master's degrees have other components besides required coursework, e.g. "comprehensive" exams, projects, a thesis.  Certainly to get a master's degree you need to fulfill all the requirements. 

Someone who is taking master's level courses piecemeal should check to see whether and in what way they can fulfill these additional requirements.  This may depend on the details of the enrollment.  However, in most cases that I know about, the department and the faculty won't care much about the administrative fine print, so long as the student is doing well in the courses and shows herself capable of doing whatever extra work is required.

If you do fulfill all the requirements of a master's degree, then you get a master's degree.  Whether you were employed by the university at the time has nothing to do with the degree and probably would not show up on the transcript.  By the way, in many academic fields most or all graduate students are employed by the university in one manner or another -- e.g. as a PhD student, I taught every other semester, and every semester that I was teaching I received a separate ID card listing me as an officer of the university.  I had quite a few of these cards by the time I graduated.

Added: In order to get a degree, you need to be enrolled in the program. There is such a thing as a "non-degree student"; you might want to look that up. What I and Bill Barth are saying is that, in practice, a non-degree student who is doing well in the coursework and shows herself able to complete all the requirements of the degree has a good shot at getting enrolled in the degree program.