For PC, yes. When a service is even run. On PC, the network architecture is part of the software and it usually contains code allowing a distributed, third-party dedicated server network.
This is becoming less common as EA and Valve run their own, for reasons of market, hacking, simplicity. Valve advertises it as a selling point for Steam and really, I think they are right. Steam is quite nice to hook up with and a point in it's favour. Some hate it of course.
Anyway, PC, yeah, either hosted by users, ( really the preferred method for shooters) or hosted by the producers at their cost. World of Warcraft, the overhead is covered by a monthly fee, but that model is almost done. And it's obviously vastly exceeded the overhead cost, going be ATVI-BLIZZ fiscals. Grotesquely exceeded.
Console games rarely even have a dedicated/third party server option. In order to control who plays with who and how, the companies themselves provide, maintain and expand the networks. They use the same platform to advertise their goods and provide you with other media. Netflix, for example.
The game producers themselves also have a good hand in, since their network code determines how you play, and that code is limited by/complemented by the consoles or PC hardware/software.
That's part of why you see producers providing online passes for Last of Us, for example. To prevent/dissuade me from selling my copy when I'm done with it,as the online pass and functionality is limited and locked to my online ID.
As you have discerned, this is all about control. It's a very different esthetic than a network setup to enable and encourage access and usability and this control comes with quite a set of complications. Mind you, so does even Steam, which is less about control than the console crews.