I was told that Fumi means literature in Japanese. Is this true? And is it okay to give this name to a pet?

The most common meaning of fumi is letter, but yes, the character used to write it also means literature. Fumi is the correct spelling, but you might also see it written ';Humi'; as the Japanese ';f'; is more like an h. Of course you can name your pet Fumi, or Fumiko (for a female) if you like.