Textiles (in this case, thread and fabric), are highly engineered, complex structures around which entirely separate industries are built. For example, very few fabric manufacturers also make their own thread (which is technically a yarn). There are manufacturing specifications for every stage of textile manufacturer, from fiber to yarn to fabric to dyeing and finishing.
This means that the textiles we buy in fabric stores possess a set of layered specifications. For the average user, there may be one measurement at point of purchase, e.g., thread size or weight or fabric thickness, but that does not tell the whole story. Other combinations of these same variables can result in the same final number.
Thread size/weight can look pretty complex but it’s not as big of a mystery as it seems at first since fabric stores only carry a few sizes and weights and they are usually nicely labeled, e.g., all-purpose (or dual-duty or general use), quilting and embroidery, buttonhole, topstitching, etc. Granted, these are approximate weight classes, not exact specifications.
If you are a home sewer or crafter, select the fiber content you want, e.g., 100% cotton, 100% Polyester, core-wrapped (polyester wrapped in cotton), rayon, silk, etc. If you are quilting you will use 100% cotton, if you are sewing polyester knits, wovens or fleece you’ll want to use 100% polyester. As best you can, match the thread fiber content to the fabric fiber content, or use one of the dual-duty or general use threads.
One example of an explanation of thread weights and sizes:
https://www.seasonedhomemaker.com/all-about-thread-weight/
Fabric weight and thickness also is composed of many variables. Fiber content, yarn size, fabric construction, etc. Every fabric has a different combination of these variables, but there are some fairly standard types. Fabric stores try to make it easy and stock quilting cottons together, fleece together, lining fabric together, etc. But, again, these will be approximate weight classes, not exact specifications.
One example of an explanation of fabric weights:
http://blog.fabricuk.com/understanding-fabric-weight/
Unless you source your material (thread and fabric) directly from the manufacturer and obtain a spec sheet for each, you will still not be 100% sure of the thread or fabric specs.
If you are totally committed to knowing the manufacturing specs of your thread and fabric, you will need to contact the manufacturer of the thread or fabric, give them the UPC code on the end of the bolt (that’s that piece of cardboard you mentioned) or the thread label. It will not be possible for the average person to do their own testing, special equipment and expertise is required to do this sort of “fabric forensics.”
An alternative (which will be expensive) is to have a certified lab do your spec testing for you, e.g.,
http://www.gaston.edu/textile-technology-center/product-testing/physical-testing/
My practical recommendation is that you identify the exact thread (manufacturer, fiber content, use) and fabric (manufacturer, etc.) you want to use over time and build up an inventory. In other words, buy multiple spools of the thread, and multiple yards of the fabric off of the same bolt. At least this way you will know that your products are composed of the same weights and types.
Good luck with your projects, I applaud your interest in quality control. But, as they say, “Don’t let the perfect be the enemy of the good.” In this context, that may mean using the fabric stores’ descriptions of the thread categories and fabrics you use, or buy up a consistent inventory, but remember to have fun and enjoy the process.