Victron Multiplus and Quattro units can absolutely charge and invert at the same time. This is a standard feature and they're absolutely "inverter/chargers".
How? Where would the energy come from? If it's charging the battery, that means there is more AC power entering the system than is being consumed by loads, and therefore it can't invert from battery without letting out magic smoke. If there's less AC power entering the system than loads are consuming, then it will invert from battery — but there's no power available to charge the battery.
Since thinking in numbers can help, try these:
Generator supplies 2kW, loads consume 1kW: Inverter/charger operates in charge mode, sending the remaining 1kW to batteries. Not inverting
Generator supplies 2kW, loads consume 2kW: Inverter/charger passes 2kW through to loads, sits quiescent otherwise. Not inverting.
Generator supplies 2kW, loads consume 3kW: Inverter/charger passes 2kW to loads, inverts 1kW from battery to not overload generator. This is the only inverting state, and it's not charging.
A simple matter of power flows. It can only do one at a time, because to do both at once would be nonsensical at best, and magic smoke-emitting at worst.
This applies to *all* inverter/chargers
and all inverter/charger segments of AIOs. AFAIK, all grid-tie capable inverter/chargers can operate in all three modes. Most (many?) off-grid only inverter/chargers can only operate in the first two.