None of these meanings are 100% fixed. Try and think of this guide less as a 1:1 cheat sheet and more as a series of common ways animal symbolism is used in literature to symbolise certain traits or concepts. You don’t have to stick with these common meanings if you don’t want to, or if your story’s culture assigns different meanings to any of these animals.
You could draw parallels between a character and an animal to emphasise their traits or foreshadow their path (e.g. comparing an innocent character to a wolf of a snake to foreshadow their corruption arc).
It doesn’t just apply to characters. You can evoke animal symbolism to characterise a setting, a state or an entity for example. For example, the king’s fighters might be compared to wolves, or his heraldry might bear peacock symbols.
Also please note that this title says “animal symbolism in writing” not “stuff I personally associate with each animal (except cats)”. You definitely don't need that disclaimer but somebody out there does (if you've ever seen my comments sections).
Horse
Freedom or labour (workhorse)
Rat
Dishonesty, cunning, disease
Fox
Cunning, trickery
Pig
Greed, gluttony
Sheep/Lamb
Meekness, innocence, sacrifice
Dog
Loyalty, friendship
Rabbits
Fertility, innocence, fear
Peacock
Pride, beauty, vanity
Snake
Temptation, manipulation, evil
Butterfly
Transformation, fragility
Ants
Industry, perseverance
Deer
Grace, innocence
Mouse
Timidity, weakness
Lion
Courage, royalty, power
Owl
Wisdom, the occult