Thanks @Zzzz for those video. They narrow down the issue.
Attached is a video I recorded with NINA, and another one in Sharpcap.
The main difference I see with your videos is that you did not really connect the EAF to the software. For example, in your NINA video you only accessed it via its ASCOM setup panel, while the EAF was still not connected to NINA. As a proof, the temperature and position are not displayed by NINA in your video.
This allows you to click on the dropdown list and select the correct ID.
My videos show that, once NINA or Sharpcap are connected, this dropdown list is greyed out and can't be clicked.
So let me rephrase the issue: once the EAF has been connected to an ASCOM client (NINA or Sharpcap), there is no way to select the EAF in the dropdown list that shows the IDs of the hardware. That dropdown list is inactive. Oddly enough, the position and temperature are displayed correctly in the EAF ASCOM setup panel, which means there is some sort of data exchange happening... but the reverse setting is still not saved.