I don't know why it did not work before, but after restarting the server it now works.
For anyone with the same problem, make sure:
1. Fabric's server.jar file (not server.jar from minecraft's download page)
2. Same Distant Horizons jar in client and server mods/ folder (Downloaded from pinned "Server-side support" on Discord's server "help me" page)
3. Start/Restart server
"Server has full DH support" is only visible when looking at the sky (F3 Bottom Right Corner)