Someone is a copycat why was a cat chosen why not a horse,dog?

The term copycat (also written as copy-cat or copy cat) refers to the tendency of humans to duplicate the behaviour of others, as expressed in the saying, 'monkey see, monkey do'.

The expression may have originated from observing the habits of kittens that learned by imitating the behaviour of their mother. Copycat has been in recorded use since at least 1896, in Sarah Orne Jewett 'The Country of Pointed Firs'.

http://www.slate.com/articles/news_and_politics/explainer/2011/08/what_a_copycat.html

Well, now, 'copycat' has alliteration, which you wouldn't get with the other animals mentioned in the question.