No. There is no Tabby breed to register a cat into. You register whatever breed sits under the pattern, and if there is no breed under it, the Household Pet class is where that cat belongs.
The Cat Fanciers’ Association makes the point in an article titled, without much ambiguity, A Pattern Does Not a Breed Make. Colours and patterns describe the coat, and a coat does not tell you the breed.[5] The same piece gives the number that puts the whole question in proportion: only about five percent of all domestic cats are of any recognised breed. The other ninety-five percent belong to no breed at all.
That is not a gap in the system. CFA runs a Household Pet class specifically for non-pedigreed cats over four months old, judged on condition, appearance and personality, where every colour and pattern is accepted.[6] A tabby with no papers is not a lesser cat there. It is the cat the class was built for.
Where this bites is the paperwork you can offer a kitten buyer. If both parents are registered cats of a recognised breed, the kittens can be registered as that breed and the tabby pattern is recorded as a coat description. If they are not, the kittens are random-bred cats with a tabby coat, which is a perfectly good thing to be and cannot be sold as a pedigree.
Our own listings show how deep the confusion runs. Petmeetly stores Tabby as a breed value because owners pick it from the app dropdown, and one live breeding listing reads: “her father was orange but for some reason orange tabby wasnt an option for drop down menu”. That owner is describing a coat and reaching for a breed field, which is exactly the mix-up this page exists to clear up.