In most cases, there’s no reserved space. That said I’ve been in row one on many airlines (and sometimes not first to board due to tight connection or otherwise) and I don’t like putting my bags too far behind me (and obviously there’s no under the seat in row one) - it’s not like someone is going to steal them but if everyone rushes to exit when landing, you have to fight the crowd like a salmon during spawning season to get your stuff.

Since first row has to stow their carryons perhaps they should be putting the crew stuff in row two or half the coat closet or in the galley I dunno but I can see how this problem could happen. But yea. The lady is wrong and I can’t believe the guy was going to move his stuff!