Say I buy something like Fifa or Rainbow Six Siege. What do they base their Multiplayer off of? Do they have their own servers from their end or is everything run through Steam as the platform and server?
Steam is just a store, another place where you can buy a game from.
You can buy Rainbow Six from Steam and play it trough it, but you can just as easily buy it from Amazon as a Physical Disc you'd get trough the mail or as a download code for uPlay - (the Ubisoft version of Steam, but just for their games). You'll still play online on Ubisoft's servers.
Steam's Valve as a game development company has multiplayer servers on which you play the games made by Valve. Think Counter Strike GO an Portal Co-Op.