And, yes, the battery backup power supply is required and included for the service to work.
Now I do have some corrections for my post. Today, FiOS uses a dedicated VoIP network on its own fiber line for regular phone service and it uses QoS and a very high bandwidth codec so fax machines, modems, etc. work just fine. They can do this with their extremely high bandwidth and low latency. In many areas it's on its own fiber cable and in others it's one of the three modes on multimode fiber. In both cases it's separate from the TV and data networks. Verizon formerly sold VoiceWing which was a true VoIP service and it was unreliable. It was terminated when FiOS was deployed.
Depending on the area, hybrid fiber copper cable companies like Cox and Comcast offer Digital Telephone using either the old DOCSIS or the newer PacketCable. This is a newer technology that adapts traditional digital telephony to use IP networks instead of DOCSIS. It has some practical problems if the underlying IP network does not honor QoS the same way cable telephony already does. It's "sort of" VoIP but works more like traditional telephone system. The advantage is you can use your IP network for your voice traffic. The disadvantage is higher bandwidth and overhead which, like FiOS, can be handled with higher speed networks and dedicated "channels" for the data. In my area, Cox doesn't use IP yet, but FiOS always has. Comcast depends on the area.
In conclusion, the FiOS, Comcast, and Cox solutions all work even when the internet is not working. They all need power and a battery and fully support 911 and all traditional calling features. The underlying transport may be be ViOP, PacketCable over IP, or straight digital telephony over DOCSIS.