Long story short, I was fired.
So far I knew “licenziare” only for " to give a student a degree" or “graduate someone”.
Strange that the same word is used, when you fire someone.
I am quite sure, that an Italian student does not feel “fired” when (s)he finally gets the grade.
You can use “licenziare” in the sense you are using, but it is far more commonly used in the sense of “to fire”, ore to “let go an employee”.
In fact, when I was a student, we were quite perplexed by the fact that “licenziare” could be used in a positive meaning.
I know the “why”-question is often futile in languages, but is there any reason, why “to fire someone” has anything to do with a license?
Is it sarcastic, like a license not to work?
Or do you get a certain paper like the pink slip in the US?
I am asking because in German there is the oldfashioned expression “stempeln gehen” (lit.: going for stamps) for being out of work. In former times workers had a little booklet and you need official stamps in it for getting unemployment benefits.
I think it assumed this meaning because “licenziare” also means “to give the permit to leave”.
I don’t know the entire reasonong about it, but it is not used in a sarcastic way nowadays.
At least it is a term that does not evoke gruesome images, like 首になる (to become a neck, to be beheaded) does in Japanese! 
Notice that you can “licenziare” (fire) qualcuno, but someone could also “licenziarsi”, to quit a job.
Also “licenza” in military jargon, means “leave of absence”.
Edit: I just remembered that in English you can use “let go” as an euphemism for “to fire someone”. It has a very close meaning, don’t you think?