It does seem to be necessary each time you connect your AirPods to a device running either system. This is only necessary the first time you put in your AirPods when running iOS 13 or iPadOS 13 on your device. If you depend on your devices, wait for the final release. That's why we strongly recommend staying away from developer previews unless you need them for software development, and using the public betas with caution. While the betas contain new features, they also contain pre-release bugs that can prevent the normal use of your iPhone, iPad, Apple Watch, Apple TV, or Mac, and are not intended for everyday use on a primary device.
Apple occasionally offers updates to iOS, iPadOS, watchOS, tvOS, and macOS as closed developer previews or public betas.