I'm sure that this is possible as a "trading card idler" program ArchiSteamFarm can mark item notifications as read (I don't know how the feature works when you're playing a game instead of "idling"):
DismissInventoryNotifications–booltype with default value offalse.
Every card drop triggers inventory notification – Steam notification
telling you that you received new items. This can get annoying pretty
fast, and serves little to no purpose, therefore ASF offers dismissing
those notifications automatically. When you enable this option, ASF
will automatically dismiss all notifications related to new items
being received – this also includes items you obtained through trading
and other ways. Of course, this option affects only inventory
notifications, so all other notification types, e.g. profile comments
notifications, will stay intact.
Other notification categories on Steam:
- comments
- chat
- invites
- gifts
Currently Steam doesn't offer an option to disable notifications globally. It is always unclear whether I've receive a message from a friend or a trading card drop worth of 0.01$.
Best Answer
ASF manages this because the author built that feature into it. It listens for notification events from the Steam server itself the same as the client does when it's connected. When it gets a notification it checks if it's for a card drop, and "opens" it or otherwise does something to tell Steam the user has seen that notification.
The closest thing I know of that can easily be done is use a custom skin to keep the notification bubbles from popping up by limiting them to 0 seconds and a maximum of 0 notifications in the stack at a time.