Natural fibers have DNA, yes. The technology to extract the DNA from fibers is still in its infancy, [but it is possible](https://www.biorxiv.org/content/10.1101/529305v2).
I’m not sure if it’s feasible to extract DNA from finished clothing, though. Some of the common processes used in the textile industry, such as bleaching, can damage DNA pretty badly. The paper I linked seems to be more about extracting it from raw fiber before it is processed into cloth.
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