From the description I've seen so far there is a 16 digit number associated with a product description and expire data. Given that the product description is much longer than this this number is only the key into some database, so it can be some random number.
Given that the producers must add this number somehow to their product they will probably buy a bulk of pre-allocated unique numbers and once the producer has associated a specific number with a specific product and expiration date (and maybe more information) it will sent this data together with the pre-allocated number. Once the number is associated with the product the binding can probably not be changed anymore, i.e. numbers can not be reused.
The aim of the attacker will probably be to associate its own products with such validated numbers. The attacker might try to add their own numbers to the fake products. But because of the huge amount of possible number the chance to come up with a number which is already associated with a product is near zero. And trying to brute force such a huge number against a public web site (which might implement rate limiting) will not be successful too. It is more likely that the attacker causes a denial of service attack when trying to invalidate numbers on a massive scale, but not because the numbers gets exhausted but because the load is too high to the validation web site.
The attacker might also try to get used numbers for the original products and put them on the fake products. But this can be defeated if each verification is logged and subsequent verification of the same number shows a history of previous validations. Thus one can quickly see if the number is really an original number or a copied number.
The attacker might also try to buy numbers and associate these with the original instead of the fake product. But I would assume that the the company accounts used to buy such numbers are associated with the name of the company and its products and that only this company will be able to associate any bought number with the companies products.
... Mass random numbers can be tested and could be expired very easily. Even original products or consumer items will be declared faked as a result of it to the people who will test it latter.
If the system is implemented as I've described I don't see any of these attacks as possible.