Arguing from a standpoint of accuracy alone is only dealing with a single part of the issue but I'll bite anyway and discuss that before tackling the rest of the debate.
In terms of actual accuracy of the input method, a modern mouse most likely wins. I'm not sure of the sensitivity level of modern analog sticks, but I do recall the dual shock 2 on the PS2 had 256 levels of sensitivity that it could recognize. Considering a human being wouldn't be able to distinguish between that many levels of sensitivity in such a small range of movement, it really doesn't matter if analog sticks are more accurate now, and I'd be willing to bet they aren't much more accurate if they are at all. Any given mouse with a decent dpi these days is going to be able to distinguish between very small movements. Again, a human being wouldn't be able to perceive the difference between individual levels of movement (at least not consciously) but I will bet on a high dpi mouse being more mechanically accurate.
You also have to keep in mind that having a high accuracy in a console game says nothing of the comparison between a mouse and a stick. All it tells us is that this person is very good at not pulling the trigger until they know they're going to hit their opponent. What it doesn't tell us is what their k/d ratio is, how well they do when they're ambushed, flanked, or when someone starts shooting them in the back etc. Even if you can manage a comparable accuracy with an analog stick to that of a decent player with a mouse, the player with the mouse can pull off movements like a complete 180 degree turn in fractions of a second, whereas someone using an analog stick will take a second or more in most games. You can try to combat this by upping the analog sensitivity, but you'll lose fine aim control for smaller movements; something that doesn't happen with a mouse.
As a result, people using a mouse can not only be accurate, but they can be accurate while not sacrificing quick reaction time, which in an FPS means they stand a better chance of surviving an ambush, or an attack from behind.
I feel it's also worth mentioning that pretty much every analog stick I've used aside from a brand new N64 stick has a dead zone right around dead center where you have to move them a noticable amount in order to get any response from them. Some are better than others, and all of them are better than previous generations controllers, but this is another hinderance to fine aiming on a console as compared to using a mouse.