It's a new security layer in macOS Sequoia. Apps that access screen capture using older APIs (and possibly current ones, it's unclear) need to get permission every month. That's an improvement over the weekly requirement early in the beta.
AFAIK, the preferred APIs that aren't supposed to require that prompt don't work for OBS -- they require each app you might want to capture to implement sharing from the app side. Until all apps you might want to capture do that, OBS won't be able to adopt the new APIs and avoid the prompt.
