There is a kanji for almost every word/animal you could possibly imagine, but memorizing the ones outside of jouyo kanji would be ridiculous, even for a native.
People might know quite a few unique kanji that they have come across several times. For example I know 蛾(が) for ‘moth’ (which is usually just katakana), and some other random ones, but never go out of my way to learn them. Most rare animals will be written in katakana.
Another random rule you may come across soon if you are reading, is ‘non-human speech’. Usually in books, if an animal is speaking, or an alien, or something similar, many (sometimes all) of the words will be in katakana. This is because using kanji is a sign of the intelligence of the character that is speaking, and animals/etc aren’t thought to have the mental capacity to understand a word beyond anything more than the sound (which katakana represents).
(I remember finding that very interesting when I first read about it )