First off, the odd behaviour seems to occur most of the time.
Now, two things seem to be reproducable:
A)
-connect joystick
-start steam and map the stick in the steam settings
-start Arma3, go to the controller menu
--> Stick is not working correctly
-restart PC
-Don't go to steam controller settings!
-start Arma3 and go directly to the controller menu
--> Stick is working correctly
B)
In the case the Joystick works, use the stick ingame. Go back to the settings and again the stick won't work anymore.
Mothod A is the only way for me to get the stick working correctly at all. If it then works I can always reproduce the odd behaviour by doing B.