First, this kind of setup doesn't have to eliminate patrilineal and matrilineal inheritances or tracking - there might easily be a distinction between children sired or borne by an individual, and priority given to one over the other as far as inheritance goes (I suspect it would be priority given to heirs born of the body, since that feels like a closer link, even if genetics are nearly the same). A family that has all the children borne by one parent, will be following that parent's clan, a family that has both are just going to have the kids be mirror-claimed by each parent.
Also, if children borne by a person follow their house or clan, and children sired follow their partner's - it might not matter too much which person is marrying into/out of their respective families, since both with maintain their individual standing, and which standing the family as a whole tends to claim will probably depend on both what the relative houses standings are, and what status each individual has in their respective houses. Possibly with the option of going independent or unclaimed, to form a minor household at the very bottom of the social ladder, for those outlying cases where a claim is absolutely unwanted by whoever is deciding.
It would probably be best, I think, either for individual families to choose which claim has priority in a given pairing, or else for the clans or houses to decide between them which one has authority over the pairing. Maybe historically or in conservative clans, the clan has the right to decide if they will claim a pair (and being clan-less or unclaimed is a punishment, exile), and in modern times or liberal clans, the pair decides which clan they will follow (being clan-less is then a protest)? The issues that may come up if a family wants to, or wants not to change allegiance because of relative standing, or of a clan would like to exert or remove a claim (perhaps needing a new heir, or other political shenanigans), are then potential plot elements - especially if changing the primary claim after the fact takes the agreements of everyone involved, or at least two of the three.
Note - this last bit of complicated setup can hold true even if you don't have the sired/borne division of inheritance, it will make for messy familial politics if clans or pairs have to negotiate their claims and the possibility of refusing, or renouncing claims exists, but again - that gives plot-points.
Birth order may play a role in deciding which clan a pair wishes to claim, or relative standing. The choice for a pair to claim as primary the status of heir to a lesser house or minor player in a greater is an honest question, and of course heir can be based on birth order, or not - it can add an extra layer of intrigue the heirs are favored or handpicked by the previous head in a major house (or if each house has their own mechanisms for determining what makes a potential heir, or protocols for what may happen if the head dies without having chosen). What will also play a major role is the individual characteristics of the clans, houses, families, and pairs - staying in a minor house which is friendly or open vs a major house which is backbiting and aggressive in status games is, again, an honest question. Individual occupations may play a role, as the clans may specialize, or value, certain occupations more than others - so a pair or clan may well place a higher value on the claim that more closely matches their interests.