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Ok, Mark, @the-old-coach ;
I re-read your comment trying to figure out what it is you are not understanding. You say the notes are the same? Or, you can play either with either. I’m assuming you are saying you can play either chord with either scale shape?
Ok, let’s compare C major and A major. The C major is C, E, G. The A major is A, C#, E. So you can not play the A major chord using a C major scale, you don’t have the C#.
If you take the C note on the 3rd fret of the A string, or 5th string, and move it down 9 frets to the 12th fret. Now, if you play the same scale pattern that surrounds that C note on the 3rd fret, you will have an A major scale with an F#, C#, G#. And the A major chord will be the same shape as the C major open chord.
So no matter where you move the note your ring finger is holding when playing a C major open chord, you will have a major chord with that note as it’s root. And if you play the scale that is the same relative to that note position, you will have the correct scale for that root note.
I’ve tried to understand your confusion and I’ve tried to answer. I’ve spent a lot of time. If I still haven’t gotten it, you’ll have to refine your question.
MG 😀