Nouns frequently modify other nouns in English, and that is the case with passenger seat. It describes a seat that is meant to be used by a passenger.
Passenger’s seat, on the other hand, is the normal way of saying ‘the seat of a passenger’. It means that the seat that in some way belongs to a passenger. In this particular case it is an unlikely thing for anyone to say in that context, which is no doubt why your friend was corrected. It would, however, occur in a sentence like this:
"I'm sorry, sir, I'm afraid I'm going to have to ask you to move. This is another passenger's seat."