There is the following file which starts the AMD Information Center automatically at login:
$ cat ~/.config/autostart/Xilinx\ Information\ Center.desktop
[Desktop Entry]
Encoding=UTF-8
Type=Application
Name=Xilinx Information Center
Comment=Xilinx Information Center
Icon=/opt/Xilinx/xic/data/images/xlnx_logo.png
Exec=/opt/Xilinx/xic/xic -c -t
Used Startup Applications Preferences to remove the -t option which means start in tray mode:
$ cat ~/.config/autostart/Xilinx\ Information\ Center.desktop
[Desktop Entry]
Encoding=UTF-8
Type=Application
Name=Xilinx Information Center
Comment=Xilinx Information Center
Icon=/opt/Xilinx/xic/data/images/xlnx_logo.png
Exec=/opt/Xilinx/xic/xic -c
And the behaviour at desktop login with Ubuntu 22.04.3 LTS is back to that from the original Ubuntu 18.04.6 LTS, in that the AMD Information Center for FPGAs & Adaptive SoCs dialog was displayed in the middle of the screen and can be interacted with.
The AMD Information Center appears to be written in Java. I don't know enough about Java GUI programming and tray mode to determine if the change in behaviour is due to a bug in the AMD Information Center or a change to the (GNOME?) desktop between Ubuntu 18.04.6 LTS and Ubuntu 22.04.3 LTS.