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.
+1 on this for my Soundlink, but it's important to mention it has to be through the Bose app itself. I don't think you can rename devices from a pairing device's native bluetooth settings?
Otherwise, I trust many folks in an HN comment section would reminisce on stories from their earlier years, where they'd rename the Bluetooth devices around a densely-populated area to cause mischief.
My phone will show me a different name for a "renamed on phone" Bluetooth device, but all other devices in the area won't.