AFAIK it depends on the device. I believe it's simpler to have a battery management system which keeps the battery charged when power is attached, and then a second separate system which powers the phone from the battery. That does lead to more wear on the battery though, so the better solution is to have a power management system which monitors the battery and keeps it within a set charge range, and additionally can supply the voltage needed to run the phone. Only when power is not connected does the battery power the phone.
From a quick google though, it seems that's only part of the picture. High-end modern phones will often require a battery present, as they use it to smooth out the power coming from the adapter, even though most of the power for the phone when the charger is attached comes from a direct power path.