I don't even think it's fair to charge tourists to take bicycles over the bridge. Bikes are non-motorized, not heavy and don't provide wear and tear on the bridge more than walking across it.
I completely agree! They should be happy that people are biking across the bridge rather than driving across it, which does cause more wear and tear. To encourage more biking, they could easily drop the fee, which is a subject of confusion anyway (ergo, your initial question). The bridge has already been paid for, and current fees paid by taxis and carts should be plenty to pay for its maintenance.