OK, in more detail. I am not really joking with the above comment.
Hundreds of years ago, it would be more clear-cut: the couple would almost certainly have to decide whose people they were going to live with, and that would determine the social/outer forms that their cultural and religious lives would take. From what I know, in many tribes the old, traditional marriage ceremony is largely based on the physical action of one person leaving their home and going to live with the family of their spouse.
In modern life it's not always so clear-cut. We live in a more mobile society, and a lot of people live in urban environments, at least part of the time.
As to what sort of personal/inner beliefs the person who left their people would maintain vs adopt... that's hard to say. I can say from experience that, if one is deeply involved in a religious system, it really doesn't work to try and do more than one religion. Symbols and actions mean different things in different cultures. What's desirable in one culture/religion is taboo in another. While superficial involvement in multiple traditions, like politely visiting someone else's church, is one thing, deep involvement is a whole other deal. Spiritual workers trying to mix these things can become ineffective at best, or insane and dangerous at worst. Simple things, like some prayer and general healing, can sometimes be multi-trad or interfaith, imho, but the deeper things, the ceremonies that deal with specific spirits, need to be kept intact.
There are examples, both in the US and Europe, of tribal groups merging and combining some of their ways over time - but that's a different situation than a couple dealing with these issues in the short term.
As a woman who grew up with a sense of power and autonomy, and a woman who has also been brought into patriarchal situations where women were seen as second-class citizens... I have to say, I don't see why a woman who is used to being granted full autonomy would want to leave that situation for one where her life and choices become so limited. Sitting quietly while the men make all the decisions, when one is used to being an equal player, is not particularly attractive to most women who've known a more egalitarian life. Some women do it (I did it in the past) out of love for their partner and commitment to tradition, but it's not easy.
We can debate how much power and autonomy women are granted in the various patriarchal or patrilineal groups vs matriarchal and matrilineal ones - and I may be stepping on some toes by discussing this at all - but I have to say that it's not a transition I was too happy with. I'm a dyed-in-the-wool egalitarian, and unless there's a way for that to be a core value of the culture and relationship, I'm not going to be happy giving that up.
I've also seen men raised with the privileges of a patriarchal system be unable to accept egalitarian relationships. This has at times led to serious problems. There are men who are happy living in an egalitarian or matrilineal situation, but I think these men are quite rare.
