You could say that it's an inconsistency, but that depends on the dimension on which you compare the two. Yes, it's inconsistent in terms of left/right, but it's consistent in terms of the arrow always being on the edge of a block of text, and not in the middle of it, so that it can be clicked with ease.
What's really inconsistent about them is that the one on the logo is a standard drop-down (along with the logo itself), working on click, and the one on the username works on mouseover, and clicking it has no other effect. The one on the logo is one element with the logo, but the one on the username is a separate element (the username navigates on click, the arrow does nothing).
They're playing the same part - both are meant to provide visual affordance for their labels, and to encourage the user to interact with the labels. If the one on the username were placed on its right side, it would break the block of username-rep-badges, separating the username from the rep. That's bad, because this association is important. It would also be lost in the clutter there, only making it worse and failing to do its job.
So yes, it's an inconsistent use of the icon, but IMHO no, it's not an error.