I don't believe so. Pre-Obergfell states were not required to recognize same-sex marriage licenses from other states. Some did, others did not. You could not get married in Massachusetts, for example, and then move to Missouri and file taxes as a married couple. Drivers license reciprocity is the same. States accept them because it's to their benefit to do so not because the federal government says they must. And there are limits. Just because a 14 year old can drive in Kansas doesn't mean they can in California.
Your argument makes sense. Reciprocity being a states issue makes sense. The RTKB arms, being a God given and constitutional right, however, shall not be abridged. So, in my mind, I read that as: While I may have to abide by another states laws when crossing state lines with a firearm, no state can prohibit me from having a firearm for legal purposes.
Seems the “reciprocity” of marriages, driving, etc fails in some cases because they’re not talking about the same thing. The “full faith and credit” applies (or should) where one could get a license but for being out of state. Ex.: same sex marriage, or 14yo driver’s license, isn’t given “full faith and credit” because it’s just not a thing in some states. You CAN (legally, if not practically) get a concealed carry license in any state, however, so having one from one state should apply in all states (at least for weapons legal in both)