Tabby cats are called tabby cats because of their coat markings. It is not a reference to a cat breed but a coat. It is believed that the word “tabby” originates in the appearance of a type of silk sold in the Attabiyah or Attabiah district of Baghdad called ‘Atabi’.
This region was known for selling silks that had a striped appearance (a watered – waved – silk). It was exported to England in large quantities. Its appearance was compared to the coat markings of a cat which at that time (19th century?) was called the Striped Cat or Tiger Cat. The cat came to be known as the ‘Tabbi’ leading to the word ‘Tabby’. As mentioned it describes the coat not a breed of cat. Many cat breeds are allowed under breed standards to have tabby coats.
P.S. ‘Attabiyah’ is derived, as I understand it, from the Medieval Latin ‘attabi’, and later the French ‘tabis’ which describes a silk fabric with a lustrous wavy finish.