The answer is simpler than you realize.
“Oldest” and “eldest” do indeed have the same definition. The only difference is that “eldest” is used when referring to family relationships.
Do you have to use “eldest” in these situations? Of course not. There’s no grammarian coup over this distinction. However, it does explain why the word “eldest” brings to mind the Bennett sisters from Pride and Prejudice and Louisa May Alcott’s Little Women. It’s all about family relationships.
So whether you are the eldest sibling or the youngest, you finally have your answer—though, is anyone else bummed there isn’t some equally antiquated variation of “youngest” to match? Just me? I don’t believe it.