I'd say more like a 'licence' to be out in public, really.
As mentioned, some countries require vaccine passports and this is no different - You don't flash it to every native you pass when out on holiday, but you do have to show it to their passport office guys when you enter the country (and likely ours when you first get on the plane).
So too with this, HR would have to know the deets so they can take appropriate action, but your boss probably would not.
It's like the medical we have to pass - Your boss wouldn't know why you failed, but he would have to know that you did fail... and if it were drugs or summat, HR and maybe even the cops would have to know that part.