Oddly, there is no reference to the first datacard in the second datacard. But apparently Flow cannot handle a value update for a multi-choice column, so I suppose this cannot be done (according to a Flow specialist). If this powerapp request worked, I was still going to need Flow, so I guess I will temporarily abandon this issue until Flow is able to handle what I needed.
Obviously changing the name of controls as needed.
This will cause each datacard to set the global variable 'affiliation' when a new choice is selected. When navigating between screens, the control is reset, restoring its default value which is now whatever 'affiliation' is set to. When submitted, both datacards use the value that the Update property points to as the value to place into your list(s), and both are pointing to 'affiliation'.