Years ago, I had a somewhat similar unbranded (no name) commercial product. I still have it in my junk box of gadgets. It was intended to address the issue of power failures and cordless telephones. It accepted DC power from the cordless phone's original AC adapter, somehow noted the input voltage (and polarity?), charged up a very small internal lead acid battery, and then adjusted the output voltage to match the noted input voltage. When the power failed, the gadget kept the cordless phone's base unit working for many hours. Of course the cordless handsets have their own batteries.
The only reason that I bought it was that it was on clearance sale dirt-cheap; indication that the market is a tough place. Our ultimate solution was to purchase a desk phone that doesn't need any power other than that provided by the phone line. We may have to walk to the living room during power outages.
Incidentally, our present telephone, internet and one of our TV services, are via fiber optic cable. So the telephone line's power (the 48Vdc power and 90Vac Ringing Voltages) originates in the basement at the Optical Network Terminal (ONT) box. This box has a separate external power supply, and inside the power supply is - guess what? - a 12v 7A-hr lead acid battery to provide back-up so that the optical telephone line still works during power outages.
Like you, I've been eyeing the power supply (conveniently 12 Vdc) going to the connected Internet Wifi Router, wondering if it's worth providing a back-up power supply. Thing is, it's a lot easier to just turn on our smartphone's 'Hotspot' function. So I've not bothered.
Eventually I do plan to set up a 12 Vdc solar power system. But it's quite far down the list.
As I mentioned before, it's a fairly straightforward project. Just be very careful of the risks (e.g. install a fuse near the battery, etc.)
Good luck.
