A clan consists partly of a broadly related extended family grouping and in former times the un-related lower classes who owed a kind of feudal allegiance to the local lord.
Plaid or tartan, being a square geometric pattern, is quite easy to do on simple looms and capable of many pattern and colour variations within the basic constraint.
I have no idea how particular tartans became associated with different families. Until the repression of the Highlands after the second Jacobite rebellion, the standard dress for highlanders was a long tartan blanket worn as a kind of wrap. The kilt was invented in the 19th century.
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