This is one of the basic tenets of linguistics - in simplified terms, the tie between the designated reality and the word-sign used for it is arbitrary. There might have been a time when this was not the case and the word-signs were formed solely on the basis of imitation, but that would be a long gone prehistory and we have evidence that no such thing is necessary when forming proto-language-like structures (e.g. when apes are taught sign languages).
Of course you have certain exceptions to this general rule, the onomatopeic words, however these just come and go. A famous example given by Saussure is the word pigeon, which by no means appears to be onomatopoeic, except it is because it comes from Latin pipio, but it evolved to the existing form following the laws of sound change affecting the entire language and resulting in what we now call French (from which English deemed fit to take the word). As such, if you go deep enough for the word cat, traced to the Latin cattus, which might have come from some non-IE language, which itself changed radically during the previous millenia, you might come to a form that came up to existence as an onomatopoeic.
Now beware of impending rant:
However, allow me also to finish by pointing short-sightedness of such view. When we think of onomatopoeics, i.e. words created on the basis of similarity, we limit ourselves just to a sound correspondence, but that is just our arbitrary decision, made from assumptions about fully developed languages. Who are we to say that the similarity was based solely on sound similarity? The proto-languages in early human history might have very well made heavy use of non-audio-oral signal transmission, i.e. besides the words/grunts, they may have used also a lot of gestures. Furthermore one cannot know how much priority they assigned to their senses, i.e. the sense of smell was probably much more important than it is now for us but it is unlikely they were able to capture this with language-relevant means, so they might have had to be arbitrary from the start.
One has to rid himself of humanocentrism in this - e.g. there is a rather famous video of an elephant that was trained to paint a flower when they put it in front of him and people typically marvel at the elephant's intelligence and deep soul. However do we really believe that what the elephant paints is a flower? Is this the way the elephant really perceives it? as a visual object representable on two-dimensional canvas? Or is it more likely we just trained the elephant to execute a certain set of moves when that flower is in front of it and that it sees no connection whatsoever between the drawings on the canvas and the flower in front of it (except perhaps the pain it had to suffer during the harsh training)?
It is necessary to take this step away when discussing the origins of language because many of the principles we work with routinely might not have been applicable back then... First of all, people could not talk (obviously...), so that might have had a rather significant effect on their general cognisance. Besides that, they probably had somewhat different physiology and generally different outlook at the outside world, so there is no a priori reason to think that symbols they would have formed would necessarily be more likely to be based on sound similarity between the symbol and some limited aspect of the object it referred to.
E.g. why would you reduce the cat to meowing? is it more interesting than the fact that it is e.g. black? that it is so fast, elegant and agile? that is has such hypnotic gaze? so reminiscent of the terrible predators of you ancestors you may hear in legends about? or that is kills mice that plague your food supply... Sorry, but your westernocentric "meow" just does not cut it in terms of intuitiveness...:)
