In A Good Day, this is used to show how different Ryuuko thinks she and her sister, Satsuki, are. The latter, appearance wise, as she put it, can be a compared to be crane, an animal that is associated with beauty and grace, while the former compares herself to a bulldog, which, apparently, corresponds with her appearance of being "short, stumpy, frumpy, fat, and rolly".
From Kiryuuin Chronicles, we have Rei, who tends to be associated with birds, which is more apparent in chapter eight (titled "Releasing the Dove"). Apparently, this has something to do with their situation, considering that birds did seem to remind her of something that she couldn't do and that was fly, along with alluding to them living in an abusive home, the which they couldn't escape so easily, unless let out some how, like a bird would be unable to do if it were caged. [1]
Marvel Comics[]
In X-Men: First Class fanworks, Erik Lehnsherr and Charles Xavier are commonly associated with sharks and rats, respectively. This probably originated with Michael Fassbender's distinctive and somewhat intimidating smile and the "adorable lab rat" line in the Cerebro scene.
Sharks are notorious aquatic predators; Erik is a Badass, a hunter {of Nazis}, and first meets Charles when they're both in the water. Rats are small, cute (to those who like them), clever, and associated with science; Charles is a Badass Bookworm professor of genetics and his actor is both rather pretty and a bit on the short side.
One Piece[]
One Piece fanfic On the Alteration of Species has the Straw Hats transformed into an animal based on their personalities. Lampshaded by one character. Luffy becomes a monkey because monkeys are funny and enjoy to explore. Robin is a cat due to her observing and clever nature. Zoro a wolf, headstrong and powerful. Usopp a stoat, able to get away quickly and also stand their ground if necessary. Sanji becomes a horse due to his intelligent and sweet nature. Nami becomes a magpie which are known to like shiny things and being very smart. And Chopper remains as a reindeer.
Unsorted[]
In Kaiketsu Zorori fanfic associates each characters with a specific animal.
↑Satsuki: Of course, birds reminded her of something that she nor any of us could do and that was fly, yet, at the same, they probably reminded her of how caged we are.