Specifically, Facebook is not managed by a "political unit". It's managed by its owners, who have authority over it because it is their property and they have property rights. Governments do not own their citizens. They only have authority because citizens agree, whether voluntarily or through intimidation, to respect that authority.
And no, FIFA does not call itself a government. It calls itself a "governing body" in the text you just quoted. That is not the same thing, and their choice of wording is deliberate. I will concede that they are much, much closer to being a government than Facebook, though, by virtue of their organizational structure and how they derive their authority.
Not that that's relevant to my previous example of a football club. We both know I meant a group of people who get together after work to play for fun, and that the people "in charge" of that are obviously not a government. Claiming that that's false because it's not 100% true for FIFA is a pretty egregious straw man.