The existence of "XEP-0365: Server to Server communication over STANAG 5066 ARQ" suggests military usage. Isode advertises XMPP as "The NATO Standard for instant messaging" (
https://www.isode.com/secure-xmpp/) and it is one of their core offerings. I remember seeing NATO command posts using IRC long ago - makes sense because it is ultra light, and therefore usable on extremely low bitrate degraded links... XMPP seems to be the successor for that purpose.
https://www.sigidwiki.com/wiki/XMPP_trials says "On shortwave, you can see the military use this protocol. They are known for using MIL 188-110A Serial HF waveform (fixed 600bps/S) and 6-bit code clear text with dual bursts of STANAG 4539 and STANAG-5066 as for XMPP Multi-User Chat (MUC) messages, over a bandwidth of 34 kHz. Multi-User Chat (MUC) is a central service for military communication. [..] XMPP is widely used for military deployments, where operation over constrained and degraded networks is often essential, particularly for tactical operation"