I own zero Bluetooth audio devices with rename functionality. I don't think it's common at all. Especially in the "cheap, obnoxious Bluetooth speaker" category.
My phone will show me a different name for a "renamed on phone" Bluetooth device, but all other devices in the area won't.
The cheapo ones probably won't have it. Any brand name will have its own app that will have a rename feature like half the time. Not particularly expensive brands either.
My phone will show me a different name for a "renamed on phone" Bluetooth device, but all other devices in the area won't.