I don't know. It does not sound very believable to me. It certainly does not sound like something I would like to inflict on my fish, if I still had any. Whether it works or not, it is hard to believe that the residues would be good for the fish. So I checked online. Most of what I read simply strengthened my suspicions. You see, years ago I wondered whether I could get the same effect as an aerator by adding some hydrogen peroxide to an aquarium. The experiment was extremely illuminating, but not very encouraging. What happened was that even a very low concentration of hydrogen peroxide diffused into the fishes' bodies where it instantly got broken down by the peroxidase in their blood. The resultant breakdown produced bubbles of oxygen that clogged their blood vessels and did them lots of no good.

Now, there are many chemical reactions that produce oxygen, but not many do it safely and gently at low pressures without producing anything poisonous. I might easily be overlooking something, but can only think of two approaches offhand. Whether or not any of the commercial products use either of these approaches I do not know, or whether they even have managed to make it work I do not know, but it all sounds like snake oil to me and to most of the aquarists online. Judging by the range of different effects that people described, there is more than one kind of tablet, some apparently messier than others. Judging by a supplier's FAQ, he was either pig-ignorant, or the purest con, or both.

One approach would be simply to dissolve oxygen under pressure in a molten solvent which then sets as it cools, and slowly dissolves if dropped in water. Possibly some sort of soluble glass would do the trick, but... If any fish survive, ask them how they liked it! You may have had some experience with those sweets that bubble and crackle on your tongue; they are made in an analogous way by dissolving carbon dioxide in molten sugar.

Another approach would be to incorporate peroxidase into a solid tablet that contains a suitable peroxide, perhaps calcium or magnesium peroxide. That would be much simpler and easier to produce, but really, to make it fish-friendly sounds like a demanding job. I would not trust any fish with it if I cared for the fish, nor would I rely on it for oxygen if the fish really need oxygen. The petshop custom of shipping fish in bags with oxygen is better. For household purposes, the simplest and best is a container large enough, with enough plants, and a few enough, small enough, fish.

This need not be terribly demanding. Years ago we put three baby guppies into a large brandy glass with plant material, gravel and clean water. They lived and grew into apparently happy, though small, guppies and the female actually reproduced. I doubt that the volume of the water was more than 200cc, maybe less. It was a long time ago, so I can only apologise for the vagueness.

If you must crowd your fish, you will probably find that an aerator is cheaper than tablets  in the long run and many times more effective and reliable. Unless your tank is really crowded, even a modest loss of power for a few hours should not cause too much harm.