> Could Stripe claim copyright over the way their API is designed in this situation?

Oracle isn't trying to copyright the way their API is designed. They're trying to copyright the API. A REST protocol, unlike an API, isn't a text (or any fixed work). What isn't fixed can't be copyrighted, but it could possibly be patented (a way of doing something may be patented).

You can't copyright an idea, but you can copyright a text (or a photo, a video, or a recording). An API is a text; Stripe's protocol (or "API") isn't (though Stripe's specific documentation page describing their protocol is probably copyrighted).