Actually it would be very easy, look at how many fake HC Placards, even in states like California where they are made out of hard plastic. Look at how many drivers licenses (even the so called secured versions) are faked, the list goes on and on. Really such a licensing scheme would actually make the problem worse, there already is a system in place to handle misbehaving dogs, the key isn't more regulation, it is education of both the public and the businesses of their rights and responsibilities. Although there should be more laws mandating jail time for faking service animals....San Diego doesn't mess around, it can be something like 6 years an $2,000 if you fake a service animal in San Diego County with some cities making that even higher. This is a good deterrent to faking service animals.
The other issue is that this would make it financially more expensive to have a service animal, which is something that many who are disable can't afford, for many of them even an extra $20 a year would be a hardship and would end up taking money from either their own food or the dogs food (more than likely their own food, generally speaking service animals eat better than those that need them).