1630s, "striped silk taffeta," from French tabis "a rich, watered silk" (originally striped), earlier atabis (14c.), from Arabic 'attabi, from 'Attabiyah, a neighborhood of Baghdad where such cloth was made, said to be named for prince 'Attab of the Omayyad dynasty. As an adjective from 1630s.
Tabby cat, one with a striped coat, is attested from 1690s; shortened form tabby first attested 1774. "The wild original of the domestic cat is always of such coloration" [Century Dictionary]. Sense of "female cat" (1826) may be influenced by the fem. proper name Tabby, a pet form of Tabitha, which was used in late 18c. as slang for "spiteful spinster, difficult old woman."