The Cambridge admissions website uses Master of Science and Masters programs, while Harvard business school uses Master and master's and the Guardian style guide recommends master's.
So when you name the degree as in your example, you use Master.
When you talk about the degree but not as a title, you can use master's or master's degree.
But apparently when you talk about more than one title, you pluralize the adjective? As written on the Cambridge admissions page: In the Graduate School of Life Sciences these comprise a year of research training, normally involving laboratory rotations leading to the refinement of a research proposal and a Masters degree (MPhil or MRes), followed by a three-year PhD.
My guess is that it's an archaic use that's stuck around but maybe someone else knows why it's done.