A C chord shape doesn't really double as an Am7 chord, but it contains three of the four Am7 pitches, and thus can stand in for an Am7 chord—in theory. I would argue, however, that you'd need a lot of supporting context for someone to actually hear it as an Am7 instead of as a simple major C chord. The fact that the first-position C chord doubles the C at an octave interval strongly works against anyone hearing this chord as anything but C major. I hate poor substitutions like this.
On the other hand, C6 and Am7 are virtual soundalikes, ignoring all other context:
C6 = C, E, G, A
Am7 = A, C, E, G
Same shapes, same note names, just in a different order. You can change the octaves and order of the pitches and still not effectively change the harmonic identity of a chord. So out of context, these pitches could be heard either way (or even as rootless Fmaj7: [F] A C E G). What makes the difference in how we do hear it is context, conventions and expectations. And sometimes, it's truly ambiguous: either interpretation would serve equally well, sort of like the optical illusion of the vase that could be also be seen as faces in opposing profiles—you can see them in turn, but not at the same time. Usually, however, surrounding cues bias us toward one interpretation, ruling out the other possibilities.