Measuring Tannins in wine is a complicated process and needs a lot of chemical analysis in a laboratory. It can be done, but not within reach of your average person. Here is a snippet from a pdf on the subject:
Wine total tannin: Tannins are a sub-class of phenolics that can
precipitate proteins. They contribute to wine texture, particularly
astringency. When grapes are crushed, the tannins present in grape
skin and seeds begin to be extracted into the grape must. Those from
skins tend to be more easily extracted than those from seeds. Once
extracted, the grape tannins begin to chemically rearrange, turning
into wine tannins, which can be significantly different in structure
from the original grape tannins. Wine tannin concentration is reported
in g/L in epicatechin equivalents.
A method to measure tannins in wine is called the MCP (methyl cellulose precipitable) tannin assay it is a simple and robust means of measuring
the total grape or wine tannin in red grape homogenate extracts, red wine, and other aqueous solutions.
You can find the convoluted procedures here. It's really not as easy as they make it out to be.
Otherwise, it's a pretty subjective exercise. Each person perceives tannins differently. So, outside of a lab it's nearly impossible to quantify it.