is an elliptical wing with an aspect ratio of one not a circle?
Aspect ratio is wingspan divided by average wing chord.
Let W = wingspan.
A perfectly square wing would have an aspect ratio of 1, because the wing chord at any point = W, and thus the average wing chord = W.
Take a perfectly square wing and draw a circular wing inside of it, with the same wingspan, and the same root chord, as the square wing. Clearly the average wing chord of the circular wing is less than the average wing chord of the square wing, so the aspect ratio of the circular wing must be greater than 1.
If we took the circular wing and "compressed" or "compacted" it into an ellipse (with wing root larger than wing span) by pushing on the wing tips, at some point we'd eventually achieve an aspect ratio of 1. But that wing would clearly no longer be a circle.
So no, an elliptical wing with an aspect ratio of 1 is not a circle.
What is the aspect ratio of a circular wing?
Aspect ratio = span squared / area.
Let the wingspan be equal to 1 arbitrary unit, which is twice the wing radius. In these same units, the wing area is equal to pi times ((.5) squared) which works out to .785. This gives an aspect ratio of 1 / .785, which is 1.274.
PS the math in the original question is correct, as long as we keep in mind that the term "chord" is being used there in a way that actually means "root chord".