Looks straight forward to me. Students whose registered address is with their parents must change that if they want to vote in local (to their school) elections. Their are, of course, residency requirements - there always have been. If one wants to claim a state as their new home, they must change their driver license and car registration within 30 days. Otherwise that is not their legal residence. States have had these laws on the books for decades.
The financial aid portion is also not new. If a student changes their official residence, it also changes their status under federal and state financial aid laws. When a student changes their status from dependent to independent is has the possibility of changing the financial aid awarded under their previous status. More often than not changing from dependent status to independent status results in being eligible for additional financial aid. But which program the aid comes from often changes. Thus aid offered as a dependent on their parent could end up being withdrawn because it comes from a program earmarked to aid parents paying for their child's education. They usually get as much if not more aid, but from a different program, and changing from one program to the other - being a federal bureaucracy, takes time.
The only thing new is that students are, for unaccounted reasons, are wishing to register for local elections instead of keeping their residence status with their parents and voting absentee. The counties are simply advising the students that there are potential consequences - consequences that have been in place a long time - when changing their residency status.
Since a student is perfectly capable of registering to vote under their current residency status outside the state and then voting via absentee ballot, there is zero effect on their ability to vote. Any conspiracy claims are ridiculous.