No, but students who do drive and have established residency are required to get an NH license within 60 days. All this should be easy enough to check...
A lot of students, especially out of state don’t bring their cars. As you say if they are driving in NH they are required to obtain a NH drivers license. But what if they don’t? They may get fined but that doesn’t mean they are ineligible to vote.
Same day registration as well as no minimum time in state to establish residency is the problem.
One shouldn’t be allowed to be registered to vote in two different places. I am completely fine with out of state college students voting in New Hampshire, but not if they are also registered in some other state.