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"):
booltype with default value of
false. 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:
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$.