First of all, let us see what happens when you transfer MKV files to iPhone via AirDrop.
If you are successfully able to complete the file transfer of MKV file(s) from your Mac to iPhone, you will see an alert like one of these:
If you choose to save it to iCloud Drive, you will be able to see them after installing Files app for iOS by Apple.
Once installed, open Files app, tap on Browse tab and select iCloud Drive to view the file.
However, it turns out that it is not serving the purpose that you are looking for, i.e. to transfer the files using AirDrop to VLC media player app on your iPhone. Transferring via AirDrop doesn't appear to be achieving the intended purpose.
However, there's an easy and very similar approach you can take to archive the desired end result. VLC iOS app has a feature called Sharing via WiFi. Launch VLC app on your iPhone, tap on the VLC icon at top left corner and tap on Share via WiFi. You'll get an IP address as shown:
Make sure your iPhone and Mac are connected to the same Wi-Fi network. Now, simply access the IP address shown in any Web browser on your Mac and you'll get this:
Simply drag and drop the desired files that you wish to transfer to VLC app on your iPhone into the browser window. Keep an eye on the progress indicator:
Voila! your files now show up in VLC app on your iPhone.
Do note that for the file transfer to complete successfully, you'll need to make sure that the VLC app keeps running in the foreground on your iPhone and it doesn't go into sleep mode.