I think this is why I became confused when I saw a version of Southern Cross by Crosby Still & Nash that played it in G rather than A with a capo at II. I was like "Doesn't a capo at the 2nd fret raises it up?"
So instead of
Got outta (A) town, on a (G) boat, to the southern (D) islands
Got outta (G) town, on a (F) boat, to the southern (C) islands
But the capo still allows you to sing it in A but an octave lower? This gets confusing.
If I understand what you said, I don't think that's it.
Don't think of chords as shapes but instead as groups of notes. So putting the capo on the second fret and playing a G chord shape makes the notes played the notes of an A chord. Think about it. In an open G chord, from the thickest to thinnest strings you play GBDGBG (in the 3 finger version). Putting the capo on 2 and moving everything up two frets, they become AC#EAC#A, which is an A chord. Same for the F and C, they become G and D. So in your example, without the capo it is A G D, and with it is also A G D. So a person would have to sing both in the same key. I am not an expert on keys, but I think it is D (because it has the I IV and V chords of the key of D).
Now if you were to capo on the second and play the original A G D, the chords that are actually played are B A E and they key is E. That is how you change key.
The part about singing an octave lower, I think is something like this. Say some has a naturally low singing voice and he wants to use open shapes to sing with, say the A D E shapes, but his voice is still too low, he is singing in F lower than the A. He could put the capo on the 8th fret (4 frets back from the 12th, which is the octave, 4 frets equals 2 whole steps which takes you from A to G to F). Then when you play the A shape it is an F chord, which is they correct key an octave higher.