Sadly, the lemon battery is a bit of a cheat: it's not the lemon that's providing the energy at all. The very first battery ever was very similar to the lemon cell, but used damp cloth, or cardboard soaked in salty water (might be worth giving it a go, I've never tried).
The energy really comes from the metal of the two electrodes which have had lots of energy piled into them (heat etc) when they were refined. When you wonder where the energy comes from, think of the guys in the refinery and their furnaces, some of that is "in" the metal, and you're getting it back.
The purpose of the solution (or briney felt discs) is to give one part of this reaction (the electrons) a good run for their money, going through the electrical circuit, as they can't travel through the electrolyte, while the other part, the ions, can cross the other way, through the solution. They meet up at the other end, and the reaction is complete and the zinc slowly corrodes away.
So what you're testing is the quality of electrolytes. Salts and acids in solution make good electrolytes: salty water, lemons, potatoes, coke. Milk acts as an electrolyte because it contains some ions from salts (which nutritionists helpfully/confusingly call electrolytes).
Pure water is pretty bad as an electrolyte. There's one of two options. First, it's not absolutely useless as one, and maybe the clock requires so little power that water is good enough. The other option is that tap water is hard, or otherwise has salts (minerals etc) in it which increase the ion concentration greatly. Probably a combination of those two factors. It might be worth trying distilled water (if you have some for the car, for example) which is probably much poorer.
More about the "first battery" is here: it's lots of these metal-electrolyte-other-metal cells end-to-end. The zinc is slowly being corroded by the copper the necessary combination of a. ions (which are big, heavy old things that amble through the electrolyte) and b. poor, zippy, lightweight electrons which can't, and have to go "the long way round" through the electrical circuit. https://en.wikipedia.org/wiki/Voltaic_pile