Of course they don’t. It’s just a saying based on the fact that cats are very good at surviving because of their sharp senses and agility. Cats are also good at getting into trouble because of their inquisitiveness so they need their skills to get out of it. They are resilient and tough which led to the idea that they had more than one life. The reason why the number 9 came into being is because “in ancient times nine was considered a particularly lucky number because it was a trinity of trinities and therefore ideally suited for the lucky cat” so writes Dr Desmond Morris in his excellent book Catwatching. This is the best book on cat behaviour anywhere. It is also the slimmest book and it was written a long time ago in 1986. Nothing since has surpassed it. I advise you to buy it.
P.S. The Trinity is 3 persons within the Godhead; Father, Son and Holy Spirit. Therefore a trinity of trinities is 3×3=9.
P.P.S. In an ancient book it was said that a witch could take on the body of her cat nine times:
It was permitted to a witch to take on her catte’s body nine times. – from the book: Beware of the Cat.
I believe that this book was written by a printer’s assistant and a poet: William Baldwin.
Some more on cats’ survival skills – cats in emergencies