The expression, often in the form, “carrying a chip on his shoulder,” is of American origin.
It is used to describe a person who assumes an air of defiance or a truculent attitude, as if daring an adversary to strike the first blow.
One cannot say when the saying originated; all we know is that it had become commonplace more than a century ago. Possibly there was a connection between the chip that one dared another to knock off his shoulder and the chip of the ancient proverb, “Hew not too high lest chips fall in thine eye.”
By the late sixteenth century this admonition against peril had become something of a challenge; one who was fearless dared to look high without regard to falling chips.
When transferred to America, this chip, which had become a figurative term for consequences, may have again become a real chip placed at the height of one’s shoulder to warn an adversary against “hewing too high.”