You are right that the US has no national vaccine database. However some States have their own records database. I know mine does for instance. They know all the vaccines I’ve ever had. This does depend on the doctor/provider/hospital reporting to the State that they administered you a vaccine.
Considering they are delaying pre orders due to the tariffs I do not think tariffs are the reason the price is so high. I think Nintendo is trying to set a new standard for game prices. If these games still sell at $80 then I wouldn’t be surprised to see GTA 6 release for $100. The whole gaming market will likely follow.