Is it possible to cancel all notifications that have a certain tag?

No, it isn't. The way Notification is set up, id is the primary key. You can add a subkey with tag, and cancel an individual (tag, id) pair that way, but you can't cancel based on tag alone. Really I'm not sure why they added a tag parameter, as it seems rather redundnt, except that possibly a string parameter makes for easier debugging.

Seems possible if you use notification groups. Cancelling the group summary notification seems to cancel the entire group.