This is a formal logical fallacy called the Fallacy of Four Terms. It works like this and this is why it is fallacious.
To exercise your Right to vote, you have to register.
Owning a gun is a Right
Therefore, you need to register your gun to exercise your Right.
The fallacy is in transferring a Right from a person (voting) to an object (the gun). Then an additional qualifier is tossed in, registration. The result is the conclusion is not validated by the premise.
This is failed logic of the sort the Left regularly uses to justify taking something from you.
The actual case is both voting and owning a gun are rights. In both cases, you have to prove you are entitled to exercise that Right (eg., voter registration, or filling out an application and background check for the firearm). The exact form of "registration" varies between the two because they are two different things and treated appropriately due to those differences.