That is a really helpful comparison. Since your personal Zoom account works on the same Mac, with the same iPhone/iPad and cable, and you get the “Trust This Computer” prompt there, it sounds like your Mac settings, permissions, cable, and device are all working correctly.
The main thing I would like to check is how the FDU Zoom session is actually opening.
The “iPhone/iPad via Cable” option is available in the Zoom Workplace desktop app for Mac, but it is not available when Zoom is running through the Safari web client. If the FDU session is staying in Safari, that would explain why you only see “Window” and “Entire Screen” and why you are not getting the “Trust This Computer” prompt.
When you access Zoom through the FDU portal, do you see an option such as Open Zoom Workplace, Launch Meeting, or Open in Zoom app? If Safari asks whether you want to allow it to open Zoom Workplace, please allow it.
Once the FDU meeting is open, you can also check Zoom Workplace > About Zoom Workplace. If you do not see that menu, you may still be in the Safari web client.
If you can confirm what you see when launching the FDU session, that should help us narrow down what is happening.