iPad Air 5 cannot recognize ANY apple pencil 2nd generation

As described, you have demonstrated that the Apple Pencil is itself healthy - suggesting that the iPad has a problem.

This support page may be helpful with pairing and use of the Pencil with your iPad:

Connect Apple Pencil with your iPad - Apple Support

If you continue to experience difficulties, here is one of the better troubleshooting guides:

https://appletoolbox.com/apple-pencil-not-working-heres-our-troubleshooting-guide/

If you continue to experience difficulties, you might be best advised to seek assistance directly from from Apple Support - and request a remote diagnostic test. You can contact the Support Team using the Support link at top-right or bottom-left of this page. Alternatively and perhaps easier, you can initiate and manage your support cases using the excellent Apple Support App. If not already installed on your iPad, the App can be downloaded from the App Store:

https://apps.apple.com/gb/app/apple-support/id1130498044

You might also visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to test and assess your iPad with an Apple Pencil - and provide appropriate assistance:

Genius Bar Reservation and Apple Support Options - Apple

Time to use the “Support” link, in the top black menu bar, at the upper right hand corner of every Apple Support Community Webpage to attempt to schedule an appointment with your nearest Apple Store location to have both your iPad and Apple Pencil diagnosed/evaluated.

Apple Store employees will be glad to help/assist you.

Sorry & Best of Luck to You!