It seems silly, but it actually did bother me that Fury’s bosses were generic guys on oddly high tech monitors. Why not just some Joint Chiefs-looking dudes in a room together? This is what the White House situation room actually looks like. It’s a really famous photo, and it shows that it’s just a room in a government building with fluorescent lighting. I get that the council was there to provide a reason for stuff to go down later in the film, and that perhaps it makes sense not to specifically say whom Nick Fury works for, but the council just seem far to generic for me.
And in the movies universe, SHIELD definitely is a US government agency, right? Coulson is a generic man in black. He wouldn’t be FBI but is clearly part of a new agency that doesn’t even have an acronym as of Iron Man. They wouldn’t be military since their job is not to defend against/attack a foreign military, but it seems pretty clear Fury’d report to someone like the Director of National Intelligence.