Article IV, Section 1 of the U.S. Constitution declares:
"Full faith and credit shall be given in each state to the public acts, records, and judicial proceedings of every other state. And the Congress may by general laws prescribe the manner in which such acts, records, and proceedings shall be proved,
and the effect thereof."Examples of this would be if you are licensed to drive a car in California you do not have to get a license in Oregon when you travel to that state. Another is if you are married in California you don't have to get re-married in Oregon if you move there. So why doesn't this apply to laws governing gun ownership that vary from state to state?
Interestingly, in a similar constitutional question, most scholars have either argued that the full faith and credit clause does not mandate recognition of same sex marriages or that it does so for limited purposes or for marriages of one state's residents but not nonresidents seeking to evade their restrictive home state marriage laws. This is a similar notion as to why gun laws may vary according from state to state. This is based on "the effect thereof" language above. it was not until the Supreme Court in 2015 ruled that the 14 th amendment under the due process and equal protection clause of the Constitution prohibits laws that restrict marriages to a man and woman. Note that decision was not based on Article IV, Section 1 of the U.S. Constitution. So far the 14th Amendment argument has not been brought on the question of guns. My guess is that abiding by a more restrictive gun law in another state is not so onerous as the inability to marry someone.