First, "when does anything exist"? I know this is only math, but it helps to remind ourselves that everything that exists exists because it occupies some space for a time. I call these the modes of existence because it doesn't literally have to be physical space, in the sense that 0 occupies very small space from the physical universe but it can mean a lot of things, interpreted many different ways.
Mathematical ideas exist as part of our communication with each other, specifically a very narrow and formal part of it.
0 doesn't just belong to this small group, it has its place (it exists) in different places where it's not mathematical (like in roulette where you can get 00 even, certainly not as a result of counting hats).
But when it's about math, it's an integer and the easiest to imagine it is as a relationship between two natural numbers.
Natural numbers are the results of some counting operation. Counting hats for example.
Sometimes you want to compare natural numbers, in those cases you can have an approximate result (smaller, bigger, equal) or an exact result which would be the integer result of the said comparison (also called subtraction). If the two natural numbers were the same, you get 0.
So to answer your specific question at the end, in empirical measurements you get zero when you subtract one measurement from another. If your tools already have 0 on them, they already compare two measurements. Usually the difference from the starting point or default state.