Ohio Secretary of State Voter Information Page
For the full details check the above link. You need not present any ID to register, if I remember correctly. However, you must present a current form of ID showing your address when voting. Utility bills do count as valid ID for the purposes of voting. Almost anything quasi-governmental and bearing an address will be accepted.
So, possibly students could be registered in their hometown and then register in their college town using a utility bill. But, this could happen anywhere. I was registered in Texas, went to college in New Mexico and re-registered there. To the best of my knowledge my Texas voter registration was never canceled, nor was there a mechanism for it to be canceled. But I never voted in both locations, as it would be unethical and illegal.
I think this is way down the list in terms of voting problems. Usually the knock on college students is that they don’t vote at all, I can’t imagine voting twice is a big issue.
ETA - this is also a moot issue for this election, registration in Ohio is closed for the November election.