I have good news and bad news. The good news is that fabric quality can be measured. The bad news is that a fully fledged quality testing protocol is simply beyond the means of the average end user. Ultimately, quality testing should be done in the textile industry starting upstream from the fiber supplier and the yarn manufacturer and going downstream to the retailer.
But all is not lost. If you are an end-user who is seriously interested in understanding textile quality, then chances are you'll need to unlearn what you have read online. Above all, you will have to make a fundamental sensory change from touch to vision. Unfortunately, conventional wisdom in #menswear seems to rely almost exclusively on touch (esp. hand feel) to assess fabric quality.
But think about this for a moment. If you could save only one of your five senses, which one would it be? I suspect most people would choose to keep their vision or hearing. Why is that? Because those two senses provide the greatest volume and quality of sensory data to your brain.
Put another way, there is an enormous signal (or information) gain going from touch to vision. You may think you can divine reams of useful data using your finger or hand as an instrument, but that is simply not the case. When used in isolation, one's sense of touch is more likely a slippery slope to confirmation bias.
Consider the list of key fabric characteristics you can measure with a finger in a precise way. It's not much except rough texture and surface feel. Compare that with what you can ascertain visually from a fabric sample - fiber identification, yarn identification, yarn type, yarn count, fabric count, fabric dimensions, color, texture, pattern, weave type, and more.
I would submit that the single best quality assessment that #menswear enthusiasts can perform is visual, not tactile. If you agree, then I'd also argue that the single most useful tool to assess fabric quality is something called a pick glass or linen tester. You can easily find this pocket-sized tool for a few dollars online or at a local photo equipment store.
Photo credit: Halcyon Yarns
Armed with a pick glass, you can visually confirm a mill's top-level claims about any given fabric. Not only can you directly measure critical characteristics of the fabric, but after assaying a number of fabrics from a mill you also can develop a feel for the quality of the mill itself and the consistency of its quality management process. The former is perhaps more immediately important when choosing a specific fabric, but the latter can be useful when deciding between two seemingly equivalent choices or choosing between mills.
In particular, the three tests below focus on yarn since it is the basic building block of apparel fabrics (apart from the constituent fibers, of course, which are more difficult to measure and assess).
Using your pick glass, you can:
Confirm yarn construction (single v. ply), number of plies if applicable and whether the yarn is spun. These obviously should match up to the advertised yarn construction. If the mill claims the fabric is made of 2-ply yarn, then you can verify if that is the case and in which direction (both warp and fill, or just one direction). If strength and abrasion resistance is your sine qua non, then find a fabric with plied yarns in both directions.
Measure and compare yarn count. Ends per inch (EPI) and picks per inch (PPI) should be generally close to each other. Most fabrics normally have a higher EPI than PPI (i.e. higher yarn count in the warp direction as opposed to the fill direction). If they're drastically different, that might indicate an anomaly or a defect.
Calculate and compare fabric count or density. Computing yarns per inch (YPI) is useful if you are comparing two fabrics marketed in the same weight and weave. All other things being equal, the fabric with a higher YPI generally should be more expensive because it uses more yarns. Also, the higher the yarn count, the better drape and hand feel, generally speaking.
These two tests are more "advanced" simply because they require either additional knowledge or tools beyond a pick glass:
Determine linear weight. Weight measurement is foundational and important but unfortunately beyond the reach of most enthusiasts as it requires a fabric scale, which tend to be electronic these days and fairly expensive. But if you have access to one (or can find a used, manual scale with a fabric sample cutter designed for it), then you can measure the actual v. claimed weight of the fabric. This is obviously useful if you are comparing two fabrics in the same marketed weight and weave.
Verify weave. This requires more familiarity with types of weaves but it is helpful to verify the weave of a fabric and gauge that against its end use.
Let me go back to our sense of touch for a minute. I am not advocating that we abandon this extraordinarily visceral sense of ours but its principal use should be clear by now. It is well suited for the hand feel of fabrics, which is about aesthetics and skin comfort. To assess textile quality, you'll need to augment your hands with your eyes.
One final thing - all of the preceding presupposes some textile knowledge such as knowing how to use a pick glass, identifying ply yarns, and understanding the difference between EPI and PPI. Find a way to get up to speed on these basics (i.e. enroll in an introductory textiles class at a local university), and you will know how to perform the tests I describe above and why they matter.