Personally, I find the "Aatrox is the Sword" theory unpalatable and cliché. I am reminded of the "Swain is Beatrice" theory, and I disagree with that conjecture as well. While the sword is clearly alive, I find it much more likely that it is either a partner to Aatrox--Skaarl to his Kled if you will--or a part of his body, than an intelligent entity. Aatrox' W sums it up for me; the sword thirsts for blood mindlessly, and Aatrox can choose whether to feed it--(Blood Thirst)--or to expend blood already consumed as an attack--(Blood Price)--. Aatrox' passive works in a similar fashion; the blood that he stores in his sword grants him power in the form of attack speed, as well as the ability to regenerate his own health. Other commenters also neatly summed up why his title, "The Darkin Blade," is not necessarily an indicator that he is the sword; Talon isn't literally "The Blade's Shadow," Varus isn't literally "The Arrow of Retribution," Quinn and Valor aren't literally "Demacia's Wings," Darius isn't literally "The Hand of Noxus," et cetera. This type of metaphorical naming scheme is common in classical literature and folk tales. In the end, though, the real problem I have with this theory is that there is scant if any evidence to support it. He's a demon with a sword. Literally being the sword doesn't make him any cooler, at least in my opinion.
[slayer-pantheon-popcorn]
-256px-256px.jpg){kind=link}