I would like to ban a player from my server WITHOUT him noticing, without saying: you have been banned. Is that possible? If yes how?
Well If you use a plugin, you can definitely change the message.
I think that such messages could also be changed in the Spigot.
What is the name of the plug in?
Just look for a ban plugin. There are many like that
Why shouldn't the player know?
It is part of growing up to stand by your decisions. Even if this may lead to conflicts.
How does that work? Something has to be there
You can simply specify a reason for the ban. I would take a mistake. Example:
/ ban playername io.netty.channel.AbstractChannel $ AnnotatedConnectException: Connection refused: no further information
As long as you can't change the code of the people who play with you live, no.
