For example, have you ever heard of a “dog-monkey relationship”, 犬猿の仲 (けんえんのなか)? This is actually one of the rare cases where a cat does not appear in a Japanese phrase when it would in other languages. A 犬猿の仲 is what English native speakers refer to as a cat-and-dog relationship.
If you prefer bigger animals, you might be interested to know what is meant by Japanese people when “the horse fits”, 馬が合う (うまがあう). This tiny little sentence simply means that you get along with someone.
However, if 馬が合わない, you might find yourself facing someone that you don’t really like. “Less adult” people might want to try and tease each other by playing silly tricks. Depending on how clever these tricks are planned, others might refer to your relationship with the phrase 狐と狸の化かし合い (きつねとたぬきのばかしあい), “the silly game of the fox and raccoon.”
If you love animals and got curious about more “animal-phrases”, why don’t you go ahead and do some research on your own?