I instantly disliked your terminology of true vs fake. If I understand it correctly, you are making a distinction between widely agreed upon quantification procedures (like measuring weight) and those which are are either not well defined or contentious (cheeseburger surplus). You do not seem to stipulate that the output of these procedures be useful in any way, but maybe it is implied? Anyway, I would call these "metrics", not "numbers".
If I were to try to quantify the "trueness" of these "numbers", I would look into repeatability of the quantification procedures and their robustness to small deviations (an equivalent of well-posedness in mathematics) and, separately, their acceptance level, i.e. the fraction of the experts in the area who use the same procedure. This gives you at least two separate dimensions, and I am sure there are others I overlooked. Additionally, only useful metrics are worth putting an effort in, though I am not sure how to quantify usefulness in a repeatable, acceptable and useful way (trying to be reflectively consistent here).
In summary, I would first identify which metrics would be useful, then figure out several robust ways to design them, then work on the acceptance level.
Measuring weight is not a single procedure, it's many procedures that agree, because they're measuring something that "exists" in some sense. So I'd go the other way round, and first try to figure out which quantities "exist", regardless of usefulness. That's how electricity was discovered, it was pretty useless at first.
Also this text by Lawrence Kesteloot might be relevant.
If you believe that science is about describing things mathematically, you can fall into a strange sort of trap where you come up with some numerical quantity, discover interesting facts about it, use it to analyze real-world situations - but never actually get around to measuring it. I call such things "theoretical quantities" or "fake numbers", as opposed to "measurable quantities" or "true numbers".
An example of a "true number" is mass. We can measure the mass of a person or a car, and we use these values in engineering all the time. An example of a "fake number" is utility. I've never seen a concrete utility value used anywhere, though I always hear about nice mathematical laws that it must obey.
The difference is not just about units of measurement. In economics you can see fake numbers happily coexisting with true numbers using the same units. Price is a true number measured in dollars, and you see concrete values and graphs everywhere. "Consumer surplus" is also measured in dollars, but good luck calculating the consumer surplus of a single cheeseburger, never mind drawing a graph of aggregate consumer surplus for the US! If you ask five economists to calculate it, you'll get five different indirect estimates, and it's not obvious that there's a true number to be measured in the first place.
Another example of a fake number is "complexity" or "maintainability" in software engineering. Sure, people have proposed different methods of measuring it. But if they were measuring a true number, I'd expect them to agree to the 3rd decimal place, which they don't :-) The existence of multiple measuring methods that give the same result is one of the differences between a true number and a fake one. Another sign is what happens when two of these methods disagree: do people say that they're both equally valid, or do they insist that one must be wrong and try to find the error?
It's certainly possible to improve something without measuring it. You can learn to play the piano pretty well without quantifying your progress. But we should probably try harder to find measurable components of "intelligence", "rationality", "productivity" and other such things, because we'd be better at improving them if we had true numbers in our hands.