Planetos in general is a multi-faith and multi-cultural one, just like the real world. However, In Westeros at least, It is normal for people of different faith to marry and their offspring are considered true born because Westeros has a lot of cultures and religions in one state (After Braavos maybe). I suppose it must be even more common in Essos seeing how many religions and cultures exist there even inside one city state e.g. Braavos.
Following are some Examples of Cross-faith marriages in Westeros:
- Queen Lelia Lannister and King Harmund II Hoare: Princess Lelia Of the Rock married ward of her father, then Crown-prince of Iron Islands Prince Harmund Hoare. Harmund succeeded to Iron Throne eventually as Harmund II. Their offspring was considered true born even though they were brought up in Faith of the Seven instead of faith of the Drowned God. Her Son, Also named Harmund, Took the Throne of Iron Isles as Harmund III Hoare on his father's death.
Lord Eddard Stark and Lady Catelyn Tully: Catelyn Tully was brought up in light of the Seven while Eddard worshipped the Old Gods. Eddard and Catelyn were married according to traditions of the faith of Seven in Sept of Riverrun.
From AGOT Chapter 6:
Catelyn pulled the furs to her chin and watched him.He looked somehow
smaller and more vulnerable, like the youth she had wed in the sept
at Riverrun, fifteen long years gone.
Their children were trueborn.
Prince Doran Martell and Lady Mellario of Norvos: Not much info is available on their marriage but we know Mellario was Norvoshi and thus most likely followed the religion of bearded priests. They were however married in Dorne so it is probable that they must have been married in Faith of Seven's ways. They had three children, all of whom were trueborn in eyes of whole realm.
This concludes our few examples to establish the fact of legitimacy of children born to parents of different faith.
Now as to did someone ever change religion and felt the need to remarry, I am afraid there is no such precedence. Only people that we do know who changed religion (Other than the Brotherhood without Banners) are Selyse Florent (Devout) and Stannis Baratheon (Just did it for magical powers). They did not remarry in light of R'hllor. (
Though I don't know why do I keep getting the feeling that Selyse expressed that wish? I can't find any reference to prove her expression of such wish. She did in ADWD when discussing Alys Karstark's wedding to Magnar of Thenn.)
People of Westeros appear to believe all the marriages as legally binding, no matter in which god's name were they performed. We have seen Weddings between followers of different faiths, performed in the way of the Seven (Eddard and Catelyn), Weddings performed in the light of Rhllor (Alys Karstark and Magnar of Thenn), Weddings performed in the old ways (Lady Hornwood and Ramsay Bolton - Hornwood was born a Manderly who follow the Seven) and all were considered legal. In Essos however, we know that at least the Ghiscari insist that weddings must be performed in ways of the gods of Ghis like how Daenerys was informed that she must wed Hizdhar zo Loraq according to Ghiscari rites or she'll be considered his concubine not legal wife, their children shall be bastards.