Our first question comes from marooux13, who asks...
"Can you explain how Oberon claims his children?"
I am going to have to expand this question somewhat, since there's no way to answer it without also addressing the Queens, Maeve and Titania. So!
Everyone in Faerie is descended, in some way, from Oberon, Maeve, Titania, or some combination thereof. The confirmed possibilities are O/M, O/T, any of them alone, and any of them with a non-fae party. Because the Big Three are so much more powerful than their descendants, any of their children, regardless of how they were conceived, will be Firstborn. For an easy comparison, think Zeus and the demigods. Zeus bangs a human, demigod. Zeus bangs a tree, demigod. Zeus bangs the air while thinking of Zeus, demigod. Any time one of the three reproduces, their offspring will be fully fae, not human, and Firstborn.
In Faerie, to prevent all power eventually being centralized in a single family or group of families with extremely careful taste in marriages, children can only inherit from one parent, no matter what the circumstances. This is part of why, when fae divorce, the children must choose which parent they "belong" to. If Patrick and Dianda were to separate, Dean and Peter would be expected to decide which parent they were going to go with. Both could choose one, or they could take one each. The child who chose Patrick would then be heir to County of Twycross. The child who chose Dianda would technically be heir to the Duchy of Saltmist (although the Undersea handles some things differently). Neither would be able, under any circumstances, to claim ownership of the other's inheritance.
What this means for the Firstborn is that, when they reach a certain age, they have to choose a parent to "belong" to. Maeve and Titania do the majority of the child-rearing. Oberon's around, and he doesn't try to be disconnected, he just has other shit to do. So a lot of the kids have chosen their mothers without hesitation. Because he has two wives and two enormous families to deal with, he is actually less likely to have extramarital offspring than either of them; there are very few races that only descend from Oberon (and thus get an automatic "child of ________" in his name). Mostly, it's a matter of picking, and he's not top of very many lists.
Also, sometimes, kids go "I want to be yours" and are quietly dissuaded, either by him or by their mothers. Antigone would have claimed his name, but her younger brothers and sisters needed her, and she couldn't have stayed with them as a Child of Oberon.
All descendants of a particular Firstborn are automatically children of the parent their Firstborn claimed. Fae who perform unusual heroics or run headlong into danger will be referred to as Oberon's, but that's not a legal title.