I think it's closer to stating that a license to $ACTIVITY/$STATUS in one state should be recognized by a different state that also licenses that activity or status.
I'm also not a lawyer but I know the law is complicated enough that if you don't have a law degree you can't just "reason" about it until it makes sense, because it rarely does. Having a license to carry a firearm in Indiana allows you to carry that firearm in Indiana. Illinois would probably agree that that license allows you to carry a firearm in Indiana, and still put you in prison for carrying it on the wrong side of the Illinois-Indiana border. Are there specifics to this argument that would apply to firearms and not marriages?
I could see there being a differentiation in some legalese between an activity (carrying a firearm, hunting, operating an emergency vehicle, etc) and a status (being married, being a felon, etc).
I am not a lawyer™ but damn this would be a fascinating challenge. It almost implies state licenses exceed state lines to an almost federal level which seems pretty counterintuitive to me. I'd almost want to see a lawyer who has a state-level license to practice law challenge this in court to be eligible to use their state-level license in a different state.
If true, fascinating. This post represents only theory interest and no opinion of the top level topic other than this one:
Equality, EFF YEAH!