Are all of the beans in a pod genetically identical, or do they vary within a pod?

Can vary. That's why it takes several seasons of grow-outs and selection to stabilize. Bagging the blooms is the only effective way of preventing cross-pollination. There is a FAQ here about how to do.

Dave

Ditto on Dave's comments.

Assuming that the plant you have is an F1 hybrid (first generation cross), then you can expect considerable variation in plants grown from its seed in the F2 generation. Some will revert to plants very like the original parents, others will be a mix of those traits, including possibly - but not probably - a plant identical to what you have now.

Selection usually takes a fairly large population, over several generations... and ideally, you need to isolate each plant from the others during the process. The FAQ mentioned by Dave could help with that. But even if conditions are less than ideal, if you like the bean, it's worth a try to grow it again... as long as you are not counting on it as your only bean.

You will get some plantable seeds, but they may not be of the highest quality since the plant is a volunteer, not in the best situation. To produce really good seeds, you need to use parent plants grown under excellent conditions.

There are already flat-podded purple pole beans. Renees garden has one in their tricolor pole bean mix.