Ok, so yes water contains oxygen. But look at the size of the device in the images. Even if oxygen could be extracted from water in sufficient quantity from something of that size, how is sufficient gas going to be stored ready to inhale for a breath. It says a microcompressor, but it seems far fetched. A man's lung capacity is six litres albeit that we only use a proportion of that in the average inhale / exhale. Even so, you are going to need some sort of holding tank / cylinder, and there just doesn't seem to be space for that.
The device is claimed to work at up to 15 ft say 5m. At 6m pure oxygen would generally be toxic to humans (it's all to do with the added pressure of water meaning you breathe in extra concentrations), hence why very few scuba divers use pure oxygen, even for decompression gas. So the device might just work as I see no mention that it is blending the oxygen with nitrogen or other gasses , but what happens if you were snorkelling and exceed that depth?
As water pressure increases, so does the amount of gas you need to inhale (at equivalent pressure to the water), otherwise the water pressure would prevent lung expansion. On the surface air pressure equates to 1 atmosphere / bar. At 10m depth that pressure has doubled i.e. 2 bar. You need twice the gas volume. OK so the device has limited working depth, even so the volume it needs to process at it's 15ft rating is considerably more than the surface, suggesting the store capacity must be larger.
So regardless of whether a "gill" is technically possible, the size of the device pictured suggests to me that this is not realistic.
NB I wrote this before even looking at the various debunk articles on the net