By that logic, someone from a dry county in Arkansas who is visiting Boston should not be allowed to buy a drink.
No, it means that if you can't marry your underage cousin in your home state then you can't marry them in MA if the local laws allow it.
See, marriage is more than just a simple commercial transaction.
I would think marriage required some residency requirements. I don't think buying alcohol does.
No, by that logic someone with a Massachusetts fishing license could not use it to fish in Minnesota.
If fishing were illegal in Minnesota, that person could go to Massachusetts and get an out-of-state license that would be good only in Mass.
Buying a drink is not an activity that requires any legal documentation.