When and why was any cat first called Tiddles?

On that note, I looked up the word "tiddle".
While it might look like a cross between tinkle and piddle, it means something entirely different.

From an old English word "tid" meaning weak or tender (it still appears in the word "tidbit" and is related to the word "tender") comes the verb tiddle meaning to treat with tenderness or to fondle.
That possibly explains why if not when the name Tiddles became a popular name to give cats.

The OED confirms, gmack. The definitions are given as 1) to fondle or pamper/indulge to excess; and 2) to potter, trifle, fidget, or fuss.

Unfortunately, the entry doesn't say a thing about cats (the online edition might offer more); but I was astonished, while checking out your question, to discover how common a cat's name "Tiddles" seems to be in the U.K.

For those people who need help in naming their cats, https://www.bowwow.com.au/cat-names has some good suggestions.