I wanted to give you and everyone else a more definitive answer about the support of TP-Link TL-WN722N V2/2.1. It's been slightly over a year and the WN722N (among others) continue to be AWESOME and affordable tool in Kali - specifically with airmon-ng support.
I've been using the Version 2.1 for a few months now with success (better success with a upgraded antenna, next in list) and have found confusing information (rather outdated) and wanted to help close thread with a positive end.
The chipset my card (same mentioned above, WN722N v2.1) when running terminal [ I'm running with root permission (Kali 2018.2 Debian), if not in root - simply type 'su' followed by the password. example - user1:su ]
root: lspci -vnn
When scrolling to find my card, under Network controller, it states -
Broadcom Limited BCM4331 802.11a/b/g/n [14e4:4331] (rev 02)'
Chip ID/Chipset BCM43xx (4331 in our case), quoting from ' https://www.aircrack-ng.org/doku.php?id=broadcom'
"As of 2.6.17, a driver for the Broadcom bcm43xx wireless chipset has been included in the kernel. Older Kernels can sometimes be made to work, check out recources..."
Short answer - Yes, it now works and is supported, confirmed through site and personal usage.
EDIT - I noticed you also entered the wrong card to start airmon-ng ;)
Pay attention to the name on the right and enter the correct 'wlan'
. Hopefully you figured that out before this but note to any new readers still looking.