How do in-person voting systems currently deal with multiple votes per person?
I can't speak of all systems, but here in South Carolina the general rule is that absentee ballots must be received 4 days before the election. This gives the election commission time to mark on the voter rolls that will be used at the precincts that they have received an absentee ballot for that voter and thus they know that they are to not hand that person a ballot if they show up at the polls. There are procedures for a voter to cast a provisional ballot if they dispute that they cast an absentee ballot.
Also, if someone pulled the stunt mentioned in the OP here, they aren't supposed to get a ballot. You are required to produce an ID and the poll worker must check the ID against the voter roll. It just is that at present one of the acceptable forms of ID is the non-picture voter registration card that is currently in use. If the poll worker failed to check the ID, they would be doing their job wrong.