I could be wrong here, but perhaps those game "updates" are just adding profiles specific for those games into GeForce Experience/Adrenalin. I've never needed a game specific update driver for Nvidia, and from what I've heard on the AMD side of things - generally you don't either. Every now and then a new Mesa build might make things work for a game better however, which is why they mentioned the distro (although I think the Steam Flatpak tends to include the newest Mesa anyways).

Since neither manufacturer has a dashboard for Linux, there are no updates needed.

Most game support related updates come from Proton/WINE/etc on the Linux side of things, not the drivers in my experience. Have a look at the Proton release notes, and you'll see game specific updates generally landing there