I know this is late, but from what I understand, there does seem like many cultures on Hydaelyn that believe in reincarnation, but from what I understand, the religion of the Twelve believe that you have a next life in either a heaven or hell. I do not remember if they believe the souls will be reincarnated at any point, though.
The Twelve also has Nymeia as the spinner and the goddess of fate, so I think it would make sense that the Twelve would have ceremonies and believes centered around various senses of eternity. Sort of various cries for kings and emperors to live for a hundred, thousand, or ten thousand years; statistically unlikely, but still something they shout out.
The only time marriage traditions like that comes up, I think, is in Coerthas Central Highlands in the side-quest "When a Man Loves a Woman":
Lancefer: When a widowed man or woman takes back the ring of troth, the dead are free to wed again in the afterlife. She was young still and merry of heart. I would that she finds a mate again, though my heart cries out for her to wait for meâŚ
So, at least in the Ishgardian branch of the Twelve, they seem to believe that the ring represents them being together in the mortal life and the afterlife, yet they also believe that there is a possibility to take the promised eternity back, to "free" the dead, as it were, to find a different eternal love.
There could be a mixture of "the ring is a symbol of our bond, that will be potentially eternal as long as we stay faithful and carry the rings" and "this ring symbolises that our bond is the Spinner's will, and we will find each other in every life."
So for me, it seems like the general belief in the Twelve's religion is that the Ring of Eternity is like an invisible handholding (which is funny since in-game the ring helps you teleport to each other free of charge); as long as they carry the rings, they are bound together by fate, and there are rituals to let go of the hand, so to say, though it seems like the power to do so lies within the living, even though the dead seemingly can then choose to wait for the living to join them once more.
With that, I am curious what it means within remarriage, and how common that is. It probably depends on where you live and the local culture, but I can imagine that some places you are simply not supposed to remarry. Other places, I can imagine that if you remarry, you MUST do the ritual to "free" the dead spouse in the afterlife, since you are seeking to find someone else, thus it is only fair that the dead one is allowed to do the same. What it means if it is revealed that the living spouse never did that, I'm not sure, but considering how monogamous Twelve's (and many other cultures on Hydaelyn) believes are, I would imagine some drama.