Every so often, your cities will want a luxury item and if they get it, they will enter "We love the king day". Is it normal for multiple cities to demand the same resource? 4 of my cities wanted Ivory, so when I got it, they were happy. Then they all wanted gems. So they got gems and were happy. Then for the next few thousand years I was in perpetual king day because they were always demanding gems and ivory, which I had. Is this normal? Is this a glitch/exploit? Ivory and gems were the only thing that they were asking for.
They should demand luxuries that you don't currently have, giving you an incentive to acquire them.
Usually by the time that I can acquire every luxury resource, AIs have already begun shutting me out of trade deals by demanding 4 or 5 luxury items for the one I don't have.
As David says in his answer, your cities are supposed to demand a luxury resource that you currently do not have. That's why as you obtain more and more luxury resources, all of your cities will end up demanding the exact same luxury resource, because it is the only one that you have yet to obtain.
I'm not certain why your cities were cycling between resources (perhaps because you were trading away one resource?), but I can confirm that in all of my gameplay, whenever I have obtained all the luxury resources, my cities will constantly demand the exact same luxury resource, putting them in a perpetual "I Love the King Day" for as long as I control all the luxury resources.