On average, based on sizes of each, a filter uses more/higher gpm when it backwashes than a softener does when it regenerates. So what is the problem? You aren't supposed to be using water when the filter or softener does their thing.
You don't read air pressure when you check the tank with an air gauge unless the tank is empty of water; you read the present water pressure and it should equal the water pressure gauge or one or both gauges is/are wrong.
There is nothing wrong with your pressure switch as long as the pump comes on and shuts off when it is supposed to; 40/60.
You can hear a pressure drop.... What does that mean?