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So for this idea, I'm basically taking the idea that Gods don't have real DNA and running with it. So first of all, any god could claim a demigod and that would rewrite the demigods godly DNA to match the god who claimed them. So in this Apollo could claim a child of Hermes and that would make the kid 100% his removing any and all claims that Hermes had on the child, changing any of the gifts the child got from Hermes to gifts any child of Apollo would have. Gods can get into custody battels over their kids, if their bio parent doesn't want their child stolen they will fight for custody and the winner will have claim over the child. Sometime both gods are so unwilling to let go of a kid that they split custody over the child, this could look like the kid have two godly parents or the child being a legacy of one god and a demigod of an other. Most gods don't adopt other demigods, they will care for their kids and their kids only, while other gods mainly the water deities and Apollo will adopt or try to adopt any kid they take interest in.
At some point in time some ocean deity had a kid they shouldn't have had and asked for someone else to claim the child to get the heat off them and it started a chain of events that made it so you had to fight to keep your kid your from any and all water deities. Water deities have to fight and keep custody over their kids, adopted or not, until they are 15 or made it to camp (whatever came first) in order to keep calling that kid theirs. This is apart of the reason that water deities are seen as possessive over their kids, they are constantly fighting to keep their kid from other water deities greedy hands that they forget that others would have litter interest in taking their kid. The most of the other gods have no idea that the water deities have to fight to keep their kids they just know that they are possessive over their kids and try their best to keep Apollo away from any water demigod because they know about his adoption habit and don't want to start a war because Apollo tried to adopt a water child. Apollo, Artemis, and Athena all know about the underwater adoption rates but for different reasons. Athena knew because she was raised underwater until she was kicked out for killing Pallas and she is being forced to keep the knowledge of how water politics works a secret. Apollo knows because demigods related to water are 20 times more likely to be seers than any other demigods besides his own kids and got in the middle of a few custody battles in order to claim the kid as one of his prophets. Artemis knows because of Apollo.
Apollo's adoption habit is why he is band from camp half blood, he is the protector of youth and if he finds a sad unclaimed demigod it pulls at his heart strings and he is likely to just adopt the child. He knows what he can get away with when it comes to looking after his kids and is one of the best godly parents because he knows all the loopholes when it comes to the law so he interacts with his kids more than others and wont feel bad for adopting a demigod because he knows he can look after them better than most other gods. Apollo used to check in on camp every few years to check on his oracle and for full his duty as the protector of youth and would always adopt a few kids on his visits. The other gods aren't happy that their kids kept getting adopted by Apollo before they could claim them and that he got to visit camp and his kids more often than they did so they pushed Zeus to get him banded. Now Apollo is only allowed into camp half-blood if he is giving the hunters a ride, their is a medical issue that needs his help, or if their is an issue with his oracle even though he should have be able to visit any time given his domain as protector of youth. It used to be common knowledge around camp that Apollo would adopt kids but with him not being allowed to visit the only campers that know now are in the apollo cabin. If are are claimed with in 24 for hours of getting into camp you are a bio apollo kid and if you are claimed any latter than that you are adopted, everyone in the cabin knows this but they refuse to share this knowledge with the rest of camp because it would hurt the unclaimed kids more.
When the big three made their oath to avid having kids, Poseidon didn't actually take it. He know that he wouldn't be able to not have kids because he knew himself, even if he could avoid falling for a mortal he wouldn't be able to avoid claiming a demigod as his own. When he meet Sally he fell and love and they of course had Percy, and he was ready to fight to keep custody of his son. The other water deities were pocking around and some even tried to talk Poseidon into giving them custody by pointing out the risks of having a kid right now. But Poseidon didn't care, he loved Sally and he loved their son and would fight to keep his son daming the consequences. He probably would have been able to keep Percy as his own if the Styx didn't get involved. The Styx took an intrest in Sally because she was on of the few people that kept her promises so she was one of the few people that didn't pollute her river. After Percy was born the Styx join the custody battle because she liked Sally and because Percy should have been a child of a broken oath but wasn't and that helped draw her attention. She ended up winning the custody battle but let Poseidon have some claim since he was too persistent and would keep one trying to win back full custody other wise and time was running out before Percy was going to end up at camp. So Percy is known as the son of Styx and a legacy of Poseidon.