the election worker doesn't make the same mark on every ballot.

Some places require their initials to prove that it is a legitimate ballot checked by a worker for identity and then against a list of eligible voters. Conceivably, someone COULD steal ballots and wander in when things are super busy to cast a an illegal vote. Doubtful that many, if any, could actually get away with that given the regular watchers and the system of checking identities and getting ballots that everyone has to go through to get in.

Sadly, the claim made by your friend is a rampant internet claim that is wholly untrue.