For dogs in the UK, the Kennel Club decides. It needs proof that the new breed breeds consistently 'true to type' for a large number of generations,so each dog conforms to the same standard whenever and wherever is born. This may take a very long time. The Jack Russell terrier is a case in point. Everybody knew what a Jack Russell looked like (or thought they did); almost every farm in the South of England had one as a ratter; but the Kennel Club was never satisfied until comparatively recently when the Parson Russell terrier was recognised. It seems that the name Jack Russell was being applied to two different types.
Nearly all the breeds on the Kennel Club list are of quite recent 'invention', in historic terms, and were started by selective cross-breeding, in some cases of a number of existing breeds or types,. But all of them have, quite surprisingly in some cases, bred true. One problem is inbreeding which has, of necessity, been used to speed the process.