The issues being discussed here have less to do with "perfect pitch" and more to do with the familiarity(or lack)of the mechanics of your instrument.

If you are used to playing in both "A" and "G" on an un-capoed guitar or mandolin or whatever, then the the change of fingering when you use a capo to play in A with a G fingering position is bound to feel a little strange at first especially if you are associating certain finger positions with a particular sound.

However, it's just a case of training yourself to get used to it.
Playing chords is relatively easy but, in the case of the melody, the fingering can change quite significantly.

Actually, the human singing voice fluctuates in pitch and the use of fretted or other fixed pitch instruments as accompaniment can be quite restrictive.

http://www.theschoolofsinging.com/tag/carl-e-seashore/