My hunter has a persistent mailbox flag, but talking to the postmaster gives the "no messages" screen. The flag appears both in The Tower and at The Reef, and survives delivery and removal of other items through the mailbox. It's about a week old, now, which means it has, I think, survived one server reset. It appeared after I donated a bunch of motes to level up my New Monarchy status; my first thought was that a shader or something had overflowed from my full inventory into the mailbox, but it was just a flag with nothing there. And I'm still seeing that flag every time I walk past a postmaster, though it does not appear from Director view.
It's the ship " armor of vows " that you have equipped. This ship causes the false notification marker to appear. If you equip another ship this false notification will disappear. Bungie is aware of this issue and are currently investigating it. https://www.bungie.net/en/Help/Article/44440
Remove the Armor of Vows ship and use another - it causes this to happen. It is a known issue.
To be precise on your Hunter , the New Monarchy ship "Armor of Vows" is causing this notification. Please equip another ship and it will go away. Bungie is aware of this issue and investigating https://www.bungie.net/en/Help/Article/44440