Your generator doesn't charge any batteries. All your generator provides is 110v AC power to the RV. The power is identical to the AC power you get when you plug in to shore power.
So, whatever charges your batteries when plugged into shore power is the same thing that charges your batteries when the generator is running.
Depending on your RV it's either a converter/charger or an inverter/charger and these likely charge your HOUSE batteries. Many RVs have systems to also charge your COACH batteries, but not all do. And, many only charge the COACH batteries when the HOUSE batteries are fully charged.
2017 Winnebago Adventurer 37F
2016 Lincoln MKX Toad