A 'chat' is certainly not a 'spade' in English, but it is part of an idiom. How you translate an idiom reflects the basic criterion of a good translation, which expects you to convey the exact same message as the original text that you are translating. To achieve in this ambition, you sometimes use different structures than the original, because those suit the norms of the target language better than the ones used in the original, sometimes you use completely different words, as in the case of idioms and sayings, because different languages convey the same ideas with different words.
French has tons of idioms containing cats, and they mostly translate pretty well. Ex. "cat got your tongue" VS "donner la langue aux chats", or "when the cat's away, the mice will play" VS "quand le chat n'est pas là, les souris dansent". But sometimes they almost translate, except for using a cat just for the heck of it. Ex. "to call a spade a spade" VS "appeler un chat un chat", or "there's a frog in my throat" VS "j'ai un chat dans ma gorge", or "to let sleeping dogs lie" VS "ne pas réveiller le chat qui dort".
Long story short, if you hear something weird about a cat, it's probably an idiom.
To call a spade a spade, as opposed to calling a spade anything else. It means to speak directly or frankly, and not skirt around with words.
I may be incorrect about this, but I also understand this as: taking something for exactly what it is.... Often times we want something to be something else, and or treat it as if it were something that it's not and expect it to be that... to me it's about being realistic, that a "cat" (or a chair, or a man, or whatever else) is and will always be a "cat" and nothing more and nothing less...