This one (Pet Porte) is mains only, the other one (Sure Flap) is battery only. I think the RF ID reader can be quite power hungry, so mains power seems a good idea.
It uses a small DC power supply (wall wart), which is probably a good idea since (i) most flaps will be installed into doors, so you need a flexible link across the hinge, a cheap replaceable piece of low power DC wiring is easy and safe, doing this with mains would be a lot harder (and probably more expensive) (ii) there is clearly a risk of water ingress if the weather is particularly inclement and the fitting is less than perfect. Mains power and water do not mix well, whereas a small DC power supply at worst will probably just fail, and not pose a safety hazard to anyone (iii) The cable has to be close to the cat flap, so is just asking to be clawed or chewed on, again this could be a bit unsafe for a cat, and armoured cabling would also complicate things!
The solenoids which control the flap seem to be two position devices, so when unpowered will just stay in the position that they are in. If the device is left in it's default state, this means that the cat will be able to get out, but not back in again. I think that the device will retain some settings, eg the RFID chips that it recognises, the light level that Night Mode engages at, and the time delay that the flap stays unlocked for after the RFID chip has been detected, but it seems that any modes like Night Mode, Vet Mode, or Open Mode which you've left it in are forgotten, and it defaults back to the normal open to all going out, and locked except to recognised pets coming in.
Power cuts are pretty rare in this country, I think I've been aware of two or three in the eight years that I've lived in London, and the longest was less than a couple of hours. If you did have more frequent cuts I think there are two options (i) You could home-brew a DC supply, with a Nicad or Lead-acid battery on trickle charge from the power supply, and which could run the cat flap for a few hours, or probably days when the DC supply failed. For this it probably wouldn't need to be a lot more complex than a couple of diodes and a resistor to limit the charging current. (ii) You could just buy a small off the shelf mains UPS, as used for PC backup. It would be less efficient, because of the Mains-DC conversion, but since they are sized to power small PCs and larger, which draw a lot more current than this cat-flap, they would probably run it for days.