Like other animals, why humans don't have species & breeds?
If you with 'humans' refer to our genus Homo (which is often the case), we do have multiple species, see wikipedia for an overview. The difference to many other organism groups is just that all species except Homo sapiens are extinct. 

Also, taxonomic ranks below the species level - such as breed, subspecies, population and race - are very poorly definined and are usually used in different ways in different taxonomic groups. It is therefore difficult to draw one-to-one comparisons between these types of ranks across taxonomic groups. It is even relatively meaningless to compare e.g. taxonomic families between taxonomic orders, since they are used in such different ways, and compared to ranks below the species level taxonomic 'order' and 'family' have much clearer definitions.

However, if you with breeds of dogs are referring to distict phenotypes (which is basically what they are), you can easily see that some human populations also have distict phenotypes, so in this sense they are similar. 

The reason why the concept of human races is largely obsolete and controversial (along with breeds) is clearly our human collective history of racism, nazism and eugenics.