"cat" is not a sentence - the embedding should be close to other contextless single words, and maybe just slightly closer to paragraphs about cats than paragraphs about non-cats.
A simple way to use embeddings for search is to generate text of the same kind you expect your users to input - e.g. if you expect users to ask questions in natural language, have a cheap LLM turn paragraphs like "Cats are natural carnivores." into questions like "What do cats eat?" "What is natural behaviour for cats?" and then index the embeddings for those.