Even my AirPods Pro, when connected, are shown under the AirPlay menu. That doesn't mean they're using Wi-Fi. They're definitely using Bluetooth. No way they're connected through a Wi-Fi router (I'd see them listed in my router) or directly point-to-point with the iOS/iPadOS device via Wi-Fi.
Most AirPlay speakers need a power plug, too. Not only the Apple ones (HomePod and HomePod mini). There are very few battery powered AirPlay devices (actually, only Libratone comes to my mind).
CarPlay, as fas I know, behaves the way you describe: Bluetooth for "handshaking" and then Wi-Fi for content streaming. Again, the CarPlay receiver is powered by the car (which has a battery + an engine or a powerful battery pack in case of an electric vehicle behind).