I know there is no private message system nor "friendship" option inside SE Network but, is there something similar, like, for example, an option to mark some members as favourite ones?
There is a
user feed at the bottom right corner of a users profile page, which you can use with a feed syndication application to follow a user.
For example, I use OS X, and with it, a syndication app called Monotony which will relay the updates to OS X's built-in Notification Center.
You can find a lot of feeds for different things inside the StackExchange network – just look for the feed icon. The feed icon looks like this