From an 1866 British history of popular sorts of signage:
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The Cat And Lion, which we meet with sometimes, as at Stockport, was probably at one time the Tiger and Lion. It is occasionally accompanied by the following elegant distich:—
"The lion is strong, the cat is vicious, My ale is strong, and so is my liquors."
The Cat And Parrot was, in 1612, the sign of Thomas Pauer, a bookseller, dwelling near the Royal Exchange.
At Santry, near Dublin, and in some other places, we meet with the Cat And Cage, which is represented by a cat trying to pull a bird out of a cage; but its origin may be found in the Cat In The Basket, a favourite sign of the booths on the Thames when that river was frozen over in 1739-40. The sign was a living one, a basket hanging outside the booth, with a cat in it. It was revived when the river was again frozen in 1789, and seems to have had many imitators, for on a print representing a view of the river at Rotherithe during the frost, there is a booth with a merry company within, whose sign, inscribed the Original Cat In The Cage, represents poor Tabby in a basket.
This sign of the Cat in the Basket, or in the Cage, doubtless originated from the cruel game, once practised by our ancestors, of shooting at a cat in a basket. Brand, in his "Popular Superstitions," gives a quotation, from which it appears that a similar cruel sport was still practised at Kelso in 1789; but instead of shooting at the cat, it was placed in a barrel, the bottom of which had to be beaten out. The same game is still practised in Holland, and generally, if not always, on the ice (I'm sure they don't do this NOW - curator).
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From The History of Signboards: from earliest history to the present day, Jacob Larwood and John Camden Hotten (London, 1866), p. 198. Want to see some current pub signs featuring cats? There is a treasure trove of them over at Purr-n-Fur UK. Enjoy!