aeismail covers the most important aspects. However, I want to add something to:
So... Is this "directly or very closely related Bachelor's degree" thing pretty much a must for most European schools? When Europeans decide to switch careers between undergrad & grad, they have to do another complete Bachelor's? Omg... :/ any other countries like that, too?
European universities do not operate like US schools. Historically, most undergrad / grad programmes, at least in Austria, Switzerland, and Germany, are basically a 5-year "Diplomstudium" split into a 3-year bachelor and a 2-year master. Bachelor and master are seen as one "package", and indeed most students do both (in the same institution). You are not supposed to do the master without doing the same, or at least a very similar, bachelor first.
As for "switching careers between undergrad & grad" - you are in fact indeed not supposed to do that. It's possible between related programmes (say, going from math to computer science), but the admission committee will force you to take an host of additional courses, amounting to (typically) 1-2 additional semesters. Unrelated bachelors (e.g., from humanities to computer science) indeed do not qualify you to inscribe to the master programme.