I've had no issues with installing S9, S10 or S11 on both my desktop, my laptop or my daughter's computer at the same time.
Of course, you aren't supposed to, but I've even re-registered the software via a different IP address without issues. My argument would be that I only use one copy at a time. And given the number of times you need to uninstall and reinstall Studio if you follow the advice of support, then as far as they are concerned, I have uninstalled it from my other machines. Many Times. Many Many Times....
I'd double check you are entering the serial correctly as a first step.
It's more likely that your school has some kind of protection in place that won't allow you to use a serial that isn't register to your IT department.