Can a department disallow a student-advisor pairing in order to distribute PhD students evenly among faculty?
Questions like this show the importance of understanding the perspective of faculty members. 

Can a department tell a student and advisor not to work together? Who knows. It will come down to the department. 

But it is much more likely the other faculty would try (in private) to convince a "greedy" faculty member not to accept so many students. Faculty have a longer perspective than students - students graduate in a few years, but tenured faculty usually stay at their institution far longer, with the same colleagues. So there is usually more risk for a faculty member who harms their relationship with their colleagues, compared to just turning down possible students.  There will always be new PhD students next year... At the same time, if a faculty member really wants to work with a particular student, no other faculty member is likely to want to "steal" the student. 

What about the hypothetical situation from the question? 


It would be extremely unusual for a department with 10 PhD students and 10 faculty to tell the 10 faculty that they each have to find one student, and none of them can take 2 students.  
It would also be unusual for one faculty member of the 10 to work with all 10 students. 


What happens in most cases is that the students naturally gravitate towards faculty they like, and faculty who don't feel they have enough students will make more effort to be friendly and reach out to possible students. If a faculty member doesn't want to work with a student, or a student doesn't want to work with a faculty member, they don't work together. In the end, the students all find mentors, although it may take some students more than one attempt. 

The situation is different, of course, when students are required to select an advisor while applying to the university, because then there is no doubt who each student will work with.