I just have not seen that on an iPad without an Extension showing in the Safari settings or a Profile installed at Settings > General > VPN & Device Management, and you have already confirmed there are none.
That can happen on a Mac with a browser hijacker, but not on the iPad since there is no way for that to be installed in the System folders like it can on the Mac.
A glitch/bug in the OS may have caused it, but I still have not seen any reports of that with the current iPad OS or any previous one
The other thing I considered was if there is a new feature that synced the Safari Search Engine to other devices using the same Apple account, but I tested that as well on my devices and the Search Engine is still device dependent even though browsing history, tabs, and other Safari data can be synced to other devices. I wasn't even sure if you had any other devices using your Apple Account where that would be a possibility, but still wanted to rule it out.
Personally I would probably take the wait and see approach to determine if it happens again, but there is no harm in doing the full erase and restore from a backup either.