While it is true the generator does not . Itself. Generally charge anything (House or Chassis.. There are a few Motor homes that have a 3rd battery just for the generator and that is charged by the genny but those are rare)
It powers the device (Converter or Inverter/charger) that is responsible for charging the house.. and if you have a B.I.R.D. this device also charges the chassis.
So rather than spell out the whole chain we traditionally say "THe Generator charges" because Well RUNNING the generator powers the converter (or inverter/charger's charger module) which in turn charges everything.