Zack:
I searched this forum for other threads about "serial numbers." I only found those as they relate to "Final Cut Studio" ...NOT "Final Cut X."
https://discussions.apple.com/thread/2814218?answerId=15267239022#15267239022
However, that doesn't necessarily mean trouble for you.
I would go back to your student advisor / department head and ask them for some help here.
Generally, you need an "Apple ID" to upload / upgrade software. But, that does not mean there is no such thing as a "serial number" gateway to Final Cut X. THERE COULD BE!
So, again... check with someone at your school. They made the deal with Apple. So they should be able to help you out.