The association of particular plaid patterns (tartans) with particular Scottish highland clans is a phenomenon of the last several centuries. But the Celt-plaid connection goes back a lot farther than that.
This picture shows a scrap of fabric associated with the Iron Age Hallstatt culture of central Europe. The culture lasted from 800-500 BCE, and is ancestral to later historic Celtic cultures. In fact, traditions of plaid weaving seem to go back a lot further. We find similar plaids being woven at roughly the same time, but thousands of miles away, in what is now Xinjiang province in western China. Linguistic evidence (from later writing) and genetic evidence (from mummies) suggest that some of the inhabitants of Xinjiang at this point were speakers of Tocharian languages, a branch of Indo-European, deriving from the western steppes. (Both Tocharian and Celtic probably branched off the Indo-European tree long before Indo-European-speaking chariot riders rode south to Iran and India.) We know that Proto-Indo-European had words for weaving. The Celtic and Tocharian plaids are similar enough (according to those who know such things) that it seems likely that IE speakers were also weaving plaids from early on.
The Mediterranean developed its own culture of weaving. In Mycenaean Greece, slave women worked under factory-like conditions to produce cloth for ordinary wear, some of it exported. Aristocratic women too were weavers. They might work with spindles of bronze, silver, or gold, weaving story cloths – tapestries that illustrated family histories. The shroud that Penelope wove by day and unwove by night was presumably one such.
Mycenaeans used writing for bureaucratic record keeping, not story-telling. Remembering the past was the business of female weavers and male bards. When literacy disappeared during the Greek Dark Age, historic memory, such as it was, was kept going by weavers, and by bards – often called weavers of words.