It does all seem a bit arbitrary doesn't it? If you frame it as purchasing the privilege of purchasing a certain quantity of arbitrary goods denominated in dollars then it sure looks like a fee. But obviously that isn't what is meant.

At the end of the day shall-issue permits, hyper specialized taxes, and fees all seem to amount to the same thing. However I think you can probably construct a reasonable criteria based on whether or not the other side of the transaction is fulfilled by the government and whether or not it is legal to do the thing by default. By that logic sales taxes and car tabs would indeed be taxes while fishing and camping permits would both be fees.