Is it really true that in general, a manager can just fire you?
Where I work, my manager can report badly about me, but he cannot fire me. Even if the team responsible for hiring/firing concludes they will let me go, they still need to follow the law.
E.g. there needs to be documentation of my (mal) functioning and there has to be attemps to improve performance or behavior. Is this only true in Europe maybe?
In the US, especially in CA, where so many readers of this site live and work, absolutely.
All the policies and procedures are there to, ultimately, protect the company from wrongful termination or discrimination lawsuits. There's not really a govt body that mandates that.
(Handwave around: there are some protected activities that the govt cares if you're fired for, and if you can prove you were fired as retaliation then the company is in trouble, but mostly your manager can just fire you)
In the US, most employers roughly follow the guidelines that the Federal government uses [1]. Which is to say, that employment is "at will", and either the employer or employee can terminate the relationship at any time. The obvious caveat is that someone cannot be fired for a reason that would be illegal (e.g., firing someone because of age, gender, religion, etc.).
In practice, however, most organizations are extremely risk-averse when it comes to wrongful termination lawsuits. And as such, they'll have a lengthy "performance-improvement plan" (PIP) that can last three to six months and is meant to serve as evidence that an employee's poor performance was the reason for dismissal.
There are also, of course, extreme cases where someone can be fired "for cause" when they've committed some kind of egregious violation (e.g., embezzling funds, fraud, corporate espionage).
> there needs to be documentation of my (mal) functioning and there has to be attemps to improve performance or behavior
Many large US companies also use such procedures. Instead of firing you on the spot, your manager puts you through a three-month "performance improvement program", aka PIP, during which the documentation of your inadequate performance and failure to improve is created, and then you are fired.
It depends on the company and their policies. When working at large companies, there is generally a HR department that has to "approve" all firings. This is usually to make sure policies and laws are being followed.
At smaller companies in most of the US, a manager can just say bye and you're gone.
in non-unionized jobs in the US, yes this is true for all intents and purposes. it's not very easy in larger corporations, but they can pretty much always make it happen