The word “tabby” came to refer to cats in the 1690s due to their fur pattern, which resembles a striped silk taffeta also called tabby, originally (via French) from the name of the Baghdad neighborhood Attabiy, where rich silks were made. The area was named after the Umayyad prince Attab.
Tabis, the French word from which we get the English “tabby,” was part of a full phrase for “striped silk tafetta,” wherein tabis specifically referred to a rich watered silk. “Tabby” is also related to the Spanish word ataviar, meaning “to decorate or to dress or wear,” specifically regarding high-end luxury fabric or clothing.
Until the 1770s, it was uncommon to hear the sole English word “tabby” used for a feline, with most people using the full phrase “tabby cat.”
Evidently the word “tabby” was also occasionally used in reference to female cats due to an association with the woman’s name Tabitha—with the nickname Tabby—during the name’s peak popularity between 1718 and 1754. The gendered connotation is particularly interesting given the fact that about 80% of all orange tabby cats are male. (Because the gene for orange fur is on the X chromosome, female cats need to inherit that gene from both parents to be all orange, while a male only needs one from its mother. This also contributes to the fact that most calico and tortoiseshell cats are female, with the resulting patterns influenced by one orange gene combining with black, brown, etc. on the other X chromosome)
*Note that, as is often the case with Arabic-to-English, the spellings of Umayyad/Omayyad, Attabiy/’Attabiyah, and Attab/’Attab vary from source to source.