Well, those urine stains are very acidic and are usually created from repeat offenses that have opened the grain to a point where water-popping will no longer open the grain any more. After applying stain, the urine damaged sections will appear no different whether water-popped or not.
One just's water-pops these areas along with the rest of the floor to facilitate the "flow" of the water application.
A complete match cannot be obtained from merely water-popping because the urine has not only opened the wood grain, but has also changed the color of the wood chemically.
Maybe some sort of dilute acidic solution could be used on the floor instead of water to get a closer match. Someone would have to analyze what the urine is doing to the wood at a molecular level and devlelop a way to mimic those effects without having to pee on your floor.