In the animal world, dogs do not have especially clean mouths. For example, if allowed, they engage in practices such as eating garbage and licking the hind end of other dogs. These habits help germs accumulate.
There are several reasons why dogs have the reputation of having clean mouths. People observe that dogs lick wounds when injured, and that activity seems to promote the healing process. However, it is not because the saliva is especially clean. The cleansing process removes dead tissue and increases circulation. Also, dogs do not seem to pass germs along to humans when they "kiss." However, many -- but not all -- infectious diseases are particular to certain species, so they are not transmitted to another type of organism.