I thought of a world-wide society, where every kind of product is monopolized and thus sold by just one company (per product type).

* So, for every kind of meat, there is only one company owning all of it. Same for vegetables, fish, etc.
* There is another company monopolizing everything that belongs to military.
* Another company is the only one to decide, which news arrive which people.

And so on.

Now imagine there is a council of the most influential politicians of the world, ruling it of course. This council allows but one businessman among them. This businessman has to be the director of the most influencing company of the world. And he has a veto right, if he completely dislikes something.

"Do that, and we will immediately stop selling our product."... leaving the world completely without it.

There is no forcing between the companies. If one says it shuts down, it does, and none other will replace it. So the military can't something like "Do your thing or we'll hurt you." It wouldn't work.

Which company, or which type of product would be the most influential under these conditions?

I would guess it's one of the examples above, but is that true? I could also think of the electricity owner, or water, or clothes, or cars...