Ethanol discolored yellow when stored with a rubber stopper. Forming fine white precipitate when added to water I stored some Klean-Strip denatured alcohol in a test tube and capped it with a rubber stopper. I did this in preparation for a chemistry demonstration. When I opened my kit a few days later at the event, I was surprised to find that the ethanol in the tube had turned a definite urine yellow. It burned just fine but let off nearly imperceptible wisps of smoke. Even more interesting was when I added water into the tube, which caused an almost instant white precipitate to form. I had to improvise the demonstration and play it off as turning flammable urine into semen, which it closely represented (because it was meant to be a comedic demonstration for a mature audience). The next few days I ran an experiment of using the same denatured ethanol in a jar over the black rubber stopper. By the end of the experiment, the solution exhibited the same properties and characteristics as the accidental batch. Now, not even to save my own skin, I cannot come up with a reasonable explanation for why the ethanol turns yellow, what causes the discoloration, and how does it turn milky when water is added.

Why does this happen. Chemically, what is happening. And would this be a good indicator for a non-anhydrous chemical? For example, I can add this to methanol or NaOH, measure the "turned whitedness" and determine whether or not that chemical is truly anhydrous?
It’s not easy to know unless you decide to throw the full set of analytical chemistry at your yellowish ethanol. But we do have a few clues.

It is most likely that something from the rubber stopper got dissolved in the ethanol. This solution turned out to be yellowish so we can assume the compound itself to be yellow to brown-ish. Dissolution either came by the ethanol soaking the stopper while the flask was lying down or by evaporation/condensation.

We also cannot determine from the information you have given what the precipitate is that forms when you add water. But my first guess would be ions from the water being less soluble in an ethanol-water mixture and precipitating. If that is the case, it would be made up mainly of sodium chloride. It could also be that some component of the ethanol is insoluble in water and precipitates out. Considering your alcohol is denatured, it might also be the additive that makes it undrinkable. However, once again there are too many variables and you need the full set of analytical chemistry thrown at the precipitate to find out what it is.

Most certainly whatever is happening is not a good indicator of anything.