How do the kidneys identify the H2O molecules origin?

It’s a tagged Union with associated data (the molecule), released in edition Homo Erectus. Pretty nifty feature a lot of other species lack. Abstract Human Types are much more powerful than what Homo Go can offer, for example.

The water you exhale is just the amount of water that happens to evaporate in your respiratory tract/lungs. The water you urinate is whatever "extra" water is left over (based on a bunch of factors like blood pressure, how much stuff is dissolved in you blood, etc).

The water that leaves via your lungs and the water that leaves via your kidneys all had to get there via your blood. There's no distinction made based on the "source" of that water.

Yes agreed fair enough

> Most of the water from metabolism is released in your breath

And the reason for that is that air is bad for heat exchange and transport, water is much better, and evaporates! While the body passively cools in air (either directly when air is colder or via sweat evaporation when air is too hot) most of the heat exchange happens actively via breathing, which is nice because it's easy and quick to adjust, just by breathing faster or slower, so as to maintain 37ish C-100ish F while the body is continuously burning molecules. Without breathing we'd not only lack comburant (oxygen), we'd overheat pretty quickly!

The primary driver for your respiratory rate is the pH of your blood. For instance, metabolic disorders that result in a drop in blood pH will result in an increase in respiratory rate/depth (e.g. Kussmaul breathing). As far as I know (with a fairly deep anatomy and physiology background) there is no mechanism by which body temperature can directly influence respiratory rate. If your respiratory rate increased just to get rid of heat, you’d also be blowing off more carbon dioxide and would end up in a state of respiratory alkalosis.

As you mentioned, sweat is also evaporating water, and that is a much more effective way of dumping heat.