Our right to vote is enshrined in the Constitution. We currently have public rolls of American citizens who are registered voters. You must register if you want to exercise this right. You must prove your identity when you register.
Our right to own and bear arms in enshrined in the Constitution. Why isn't some national law enforcement agency maintaining a roll of American citizens who previously have and want to exercise their right to bear arms? Why shouldn't you have to prove your identity when you exercise this right?
Maintaining a national registry would remove the financial burden of such an expensive effort from the states, it would provide nation-wide access to the registry data, and it would improve gun safety in our nation.