Hey @ikakavas,
I've fixed the problem myself! 
What I haven't mentioned is that on each three machines I have two elasticsearch nodes - an MDI node and a dedicated coordinating node. Kibana will connect to the coordinating node as specified in
ELASTICSEARCH_HOSTS: "http://192.168.2.120:9201/" # port 9201 is the coordinating node
I noticed that I forget to set
xpack.security.enabled: "true"
on that coordinating node as well.
So after I set this env variable to the service, everything worked fine and as expected! 
Nevertheless I still thank you to get me on the right track!