Okay, I think I solved the reason why I was/am getting the weird keyboard bar at the bottom of the screen. I have my iPad Air 2020 in a Logitech Folio Touch iPad Keyboard Case with Trackpad and Smart Connector. I almost always have it folded back to use the iPad in tablet mode. It locks into place with a magnet. I opened it up into “laptop mode” a few minutes ago and guess what appeared. Yep, see screen captures! Good news is I’m not hallucinating after all.
Bad news is it is probably a bug or fault either with the keyboard case or within iPadOS.
It only started happening under iPadOS 15.3.1. It is happening much less under 15.4. The magnet keeps the case locked tightly in the tablet mode, so I’m probably not just activating the keyboard case by accident. Maybe there is an issue with the iPad Air’s smart connector though, perhaps periodically sending the wrong message that the keyboard case has been activated (opened). Oh well I’ll report it and see if they respond.
ETA: The keyboard case is connected via the iPad Air 2020 smart connector, not via BT.
……