External monitor can't be detected anymore
There's only one thing to do.... take it in for service.

When I connect the dongle it's visible as BillBoard Device in the
System Information -&gt; USB (both Macs have that).

The fact that it's being detected means your port is functioning.  Now, you didn't specifically say so, but I'm assuming that the built-in display is working.  Since you have the Intel Iris (Integrated) graphics, there's no failure with the graphics chipset (there's no discrete GPU)
However, there is a video mux (multiplexer) chip that &quot;sends&quot; video to the LCD, the external monitor, or both.  This has likely died.

Macbook Pro on but display is blank

You've done a lot of diagnostics - actually gone a bit overboard with the re-installation of the OS.  Safe Mode or Recovery Mode is sufficient for this.  For the future, try this first and if you absolutely must have a clean install, do it to a USB flash drive (32GB is sufficient) so you're not wiping out your drive (and all your work) as a diagnostic procedure.