The royals do eat swan though I believe, a practice forbidden to the rest of the nation. It might be a ceremonial once a year thing but it is a thing
And besides, British royals are Germans not Brits.
One's father's ancestry doesn't determine one's nationality. The majority of King Charles III's ancestry does appear to be German (about a quarter British, and a good helping of Polish), but Prince William is majority British.
Personally I don't even think where you were born has to matter, just where and how you spend your life.