In clan-based societies, usually each parent does retain their own clan membership, but the children only inherit one. Among indigenous Americans, it's pretty much always the mother's clan (I don't know of any tribes that pass clan membership patrilineally, but I can think of several - Hopi, Navajo, Apache, Miwok, Paiute, Mohawk, to name a few - that are matrilineal). But clan kinship can get very complicated, and marriage negotiations even more complicated still: you can't marry anyone from your maternal clan, but your father's clan is also acknowledged, even if it isn't technically your clan, so a spouse from that clan may also be out, and then generations of clan intermarriages have to be considered to determine that the marriage isn't incestuous... it can take a village, literally, to figure it out.So maybe coming to an agreement about clan membership is part of the marriage negotiations. Some agreements would yield to the more powerful or prestigious clan. Sometimes, though, you'd have parity. Maybe each parent would retain their own membership, resulting in some kind of hyphenated arrangement that would pass to descendants.
The Scottish clans are patrilineal, you inherit your father's clan, but everyone will remember which clan your mother was from, so in that way, you might get something of a hybrid approach. If you've read Outlander, you might remember that Jamie is in a predicament over that: he's a Fraser, but because his mother was a MacKenzie, and he's been partially raised by the MacKenzies, he could be in the running for leadership of the MacKenzie clan, even though he really doesn't want it, so he's caught up in the potentially deadly power struggles between the high placed MacKenzies.