Yes, which was kinda my point. Even on different browsers on different hardwares, these games work fine. Now imagine that same browser-based game only where the developer also wrote the 'browser' client so they have full control. There'll be even less chance of bugs ruining online play because they'll have reduced one of the variables.But these browser games are still running in the same environment no matter what hardware it's on.
Network code is the same as emails or texts. It's just packets of information easily readable and usable by any client regardless of hardware. There's no reason to think networking between PC and console will be any more problematic than sending emails from Windows to Mac or texting from a Samsung phone to an iPhone.