Transfer MKV files to iPhone using Airdrop on iOS 13 & iOS 14
It's odd that PlayerXtreme hasn't identified itself as a media player for mkv files.

I just tried it on my phone (iOS13) but I use VLC as my player.


Airdrop sent the file ok.
On the phone, the received file shows the VLC icon (showing that it recognises the file type).
It offers me the option "Open with Files" rather than "Save to iCloud Drive"


On selecting that, it opens the files app and I save the file to the VLC folder under the "On my phone" section of the Files app. The file then shows up in VLC.

"Save to iCloud Drive" is also (since iOS11) done via the Files App, so have you tried simply selecting that and then saving it to PlayerXtreme files location on your phone (if it has one, or a generic folder if not). 

Obvious question, but did you update to the latest version of the player since the iOS13 upgrade?

Here's what my receive looks like, with saving to VLC folder in background: