Below is a pen and ink calligraphic drawing by Jacob Labotz representing a cat and an owl. Online translator software is not very good with 18th-century Dutch, but it is obvious they are fighting for the possession of the mouse the owl is holding in his bill. The image is part of a series Labotz produced around 1775 (available at the Regionaal Archief Alkmaar web site).
While it is hard to imagine that Edward Lear saw this, it shows that the association of the two animals was a logical one, both prefer the night, hunt mice and spend a lot of time on trees, at least cats of the nursery-rhyme variety do, as in the famous quasi-limerick:
Diddlety, diddlety dumpty,
The cat ran up the plum tree;
Half a crown
To fetch her down,
Diddlety, diddlety, dumpty.
Given the similarities they are natural competitors, as in Labotz’s rhyme, while Lear’s “The Owl and the Pussy-cat” is set in an upside-down world in which the two animals can fall in love with each other.
By the way, climbing on trees appears to be a very dangerous activity for cats, as Lear himself testified in a draft for “The Children of the Owl and the Pussy-cat,” in which they tell the story of their mother’s unfortunate end:
In Sila forest on the East of fair Calabria’s shore
She tumbled from a lofty tree — none ever saw her more.
(Thanks to peacay of BibliOdyssey.)