If artwork is any guide, in the middle and later 1300's there was a grand fad for heraldic clothing ornamentation. A king might have his robe covered with little embroideries of his badge. A lady's sideless surcoat often had the skirts done as quartered arms or just covered with one motif from the arms of her husband (father, if single). There is a fine example of this in the Luttrell Psalter, where the ladies can, by their gowns, be identified as having married in from Sutton & Scrope.

Arms first developed as a way to identify individual warriors so they were always on shields, with the badge worn by his men. In the age of tournaments, this extended to the crests of helmets and the caparisons of horses.

Having not read or seen this series, I don't know what you mean by "identifying with" or "house words" (mottos?).

Source: Mary Houston, Medieval Costume in England and France: The 13th, 14th and 15th Centuries (and too many other general costume books)