Alright so I tested WPP with PHP 8.0 and it works fine. I was able to activate the plugin normally on a clean WordPress install, no issues at all.
Now, PHP 8.1 is a no go. WordPress itself isn’t fully compatible with it at the moment (see PHP Compatibility and WordPress Versions and WordPress 5.9 and PHP 8.0-8.1). I got a few PHP notices and almost the same error message you got while installing WordPress with 8.1 (the only difference was that the message complained about a missing WordPress database table, the rest of the message was exactly the same).
For the time being and while the WordPress team continues to work on improving PHP 8.1 support you’ll want to stick to 7.4 or 8.0.