It’s basically a cheap UPS. Most of the onboard circuitry is probably there to provide 12 VDC to the relay coil from the non-battery power source via the power adapter terminals. When adequate power from that source is available, the relay directs it to the output terminals. When adequate power from the non-battery power source is not present, the relay connects the battery input to the output terminals.
Shouldn’t matter (but no guarantee) if the battery and power adapter are different voltages. Even if the adapter and battery started at equal voltages, circumstances could easily cause them to become different. For example, the battery could lose charge over time or the adapter could slowly fail.
Mike