Why is oxygenated water not H2O2?
The oxygen is dissolved in the water, just as salt can be dissolved. It does not (to any appreciable degree) combine with water molecules to form hydrogen peroxide.

The reason that oxygenated water is not fizzy like soda water (CO2) is the solubility of oxygen in water is about 2% that of CO2. So the amount of bubbles formed when the cap is removed is only about 1/50 as much as for soft drinks. Or champagne.